The number ${\log }_{2}7$ is

Important Questions on Linear, Quadratic, and Exponential Models

EASY

Mathematics>Functions>Linear, Quadratic, and Exponential Models>Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another

If ${\log }_{10}x+{\log }_{10}{x}^2=2{\log }_{10}x+1$, then what is the value of $x$?

MEDIUM

Mathematics>Functions>Linear, Quadratic, and Exponential Models>Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another

If ${\log }_5{\log }_8\left(x^2-1\right)=0$, then a possible value of $\mathrm{x}$ is

EASY

Mathematics>Functions>Linear, Quadratic, and Exponential Models>Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another

If $5^{x-1}={\left(2.5\right)}^{{{\log }_{10}}^5}$, then what is the value of $x$?

MEDIUM

Mathematics>Functions>Linear, Quadratic, and Exponential Models>Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another

There are $n$ zeros appearing immediately after the decimal point in the value of ${\left(0.2\right)}^{25}$. It is given that the value of ${\log }_{10}2=0.30103$. The value of $n$ is

MEDIUM

Mathematics>Functions>Linear, Quadratic, and Exponential Models>Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another

The inequality ${\log }_a\left\{f\right(x\left)\right\}<{\log }_a\left\{g\right(x\left)\right\}$ implies that

EASY

Mathematics>Functions>Linear, Quadratic, and Exponential Models>Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another

If $\log x=1.2500 \text{and} y=x^{\log x}$, then what is logy equal to?

HARD

Mathematics>Functions>Linear, Quadratic, and Exponential Models>Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another

The value of $0.{04}^{{\log }_{\sqrt{5}}\left(\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots \right)}$ is

EASY

Mathematics>Functions>Linear, Quadratic, and Exponential Models>Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another

If $S=\frac{1}{2}\left(\frac{1}{2}+\frac{1}{3}\right)-\frac{1}{4}\left(\frac{1}{2^2}+\frac{1}{3^2}\right)+\frac{1}{6}\left(\frac{1}{2^3}+\frac{1}{3^3}\right)\dots \dots \dots$ then $S=$

EASY

Mathematics>Functions>Linear, Quadratic, and Exponential Models>Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another

The value of ${\left({\log }_{3}30\right)}^{-1}+{\left({\log }_{4}900\right)}^{-1}+{\left({\log }_{5}30\right)}^{-1}$ is

MEDIUM

Mathematics>Functions>Linear, Quadratic, and Exponential Models>Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another

What is the solution of the equation $x{\log }_{10}\left(\frac{10}{3}\right)+{\log }_{10}3={\log }_{10}(2+3^x)+x$?

EASY

Mathematics>Functions>Linear, Quadratic, and Exponential Models>Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another

What is the Value of ${\log }_{10}\left(\cos \mathrm{\theta }\right)+{\log }_{10}\left(\sin \mathrm{\theta }\right)+{\log }_{10}\left(\tan \mathrm{\theta }\right)+{\log }_{10}\left(\cot \mathrm{\theta }\right)+{\log }_{10}\left(\sec \mathrm{\theta }\right)+{\log }_{10}\left(\mathrm{cosec} \mathrm{\theta }\right)$?

EASY

Mathematics>Functions>Linear, Quadratic, and Exponential Models>Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another

If ${\log }_{4}x+{\log }_{16}x+{\log }_{64}x+{\log }_{256}x=\frac{25}{6}$ then the value of $x$ is

MEDIUM

Mathematics>Functions>Linear, Quadratic, and Exponential Models>Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another

Consider the following statements:
1. ${\log }_{10} 50$ is a rational number.
2. ${\log }_{100} 10$ is an irrational number.
Which of the statememnts given above is/are correct? 

EASY

Mathematics>Functions>Linear, Quadratic, and Exponential Models>Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another

What is ${\log }_{10}31.25$ equal to?

HARD

Mathematics>Functions>Linear, Quadratic, and Exponential Models>Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another

If ${\mathrm{3}}^x=4^{x-1}$, then $x =$

EASY

Mathematics>Functions>Linear, Quadratic, and Exponential Models>Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another

The logarithm of $\frac{1}{256}$ to the base $2\sqrt{2}$ is

MEDIUM

Mathematics>Functions>Linear, Quadratic, and Exponential Models>Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another

The value of ${\left({\left({\log }_{2}9\right)}^2\right)}^{\frac{1}{{\log }_2\left({\log }_{2}9\right)}}\times {\left(\sqrt{7}\right)}^{\frac{1}{{\log }_{4}7}}$ is

MEDIUM

Mathematics>Functions>Linear, Quadratic, and Exponential Models>Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another

The inequality ${\log }_2\frac{3x-1}{2-x}<1$ holds true for

HARD

Mathematics>Functions>Linear, Quadratic, and Exponential Models>Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another

Let $f$ be a function defined on the set of all positive integers such that $f\left(xy\right)=f\left(x\right)+f\left(y\right)$ for all positive integers $x, y.$ If $f\left(12\right)=24$ and $f\left(8\right)=15$ , the value of $f\left(48\right)$ is

HARD

Mathematics>Functions>Linear, Quadratic, and Exponential Models>Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another

Let $S$ be the sum of the digits of the number ${15}^2\times 5^{18}$ in base $10$. Then,