Logarithmic Function and its Properties

If ${\log }_{10}2=a \mathrm{and} {\log }_{2}3=b$, then express ${\log }_{4}100-2{\log }_{27}100$ in terms of $a$ and $b$.

Find the value of $x$ when $3^x=5$ 

(Use ${\log }_{10}3=0.4771$ and ${\log }_{10}5=0.6989$)

A controlled experiment took place in a laboratory. Biologists estimated the number of microbes $N$ present in a culture $t$ days after the start of their experiment. The first results obtained are shown in the top two rows of the table below.

$\begin{array}{|lccccc|}\hline \mathit{t}& 3& 5& 10& 15& 20\\ \mathit{N}& 115& 178& 480& 900& 1600\\ \log N& & & & & \\ \hline\end{array}$

Complete the last row of the table, giving values to $2$ decimal places.

Draw a graph of $\log N$ against $t$, using appropriate  scales.

Find the value of $x$ when $\ln \left(x\right)=4.2$ 

(Use $e^{\frac{21}{5}}=66.6863$)

Given $\ln 27=3.2958$, find the value of $0.5+\ln 3$.

Find the value of $x$ when $\log \left(x\right)=3$.

Find the value of $x$ when $\log x=1$.

Given $\log 27=1.4314$, find the value of $2+\log 3$.

If $\frac{\log x}{b-c}=\frac{\log y}{c-a}=\frac{\log z}{a-b},$ then

If $\frac{\log x}{b-c}=\frac{\log y}{c-a}=\frac{\log z}{a-b}$ , then which of the following is true?

Let $f\left(x\right)=\log \mathrm{ }\left(\frac{2\mathrm{x}-1}{2{\mathrm{x}}^3+3{\mathrm{x}}^2+\mathrm{x}}\right)$, then domain of $f\left(x\right)$ contain the set

$\sum _{\mathrm{n}=1}^{\mathrm{n}}\frac{1}{{\log }_{2^{\mathrm{n}}}\left(\mathrm{a}\right)}=$

The sum of the series${\log }_{4}2-{\log }_{8}2+{\log }_{16}2 -...$ is