Name: Solving Exponential and Logarithmic Equations
Uploaded: 2019-07-19
Duration: 10 min 51 s
Description: In this video, we will learn the exponential and logarithmic equations and method of solving the exponential and logarithmic equations in a detailed manner.

Question

The variables $x$ and $y$ satisfy the equation $y = a \times x^{n}$ , where $a$ and $n$ are constants. The graph of $\ln y$ against $\ln x$ is a straight line passing through the points $(0.31, 4.02)$ and $(1.83, 3.22)$ as shown in the diagram. Find the value of $a$ and the value of $n$ correct to $2$ significant figures.

Accepted Answer

Given the variables $x$ and $y$ satisfy the equation $y = a \times x^{n}$ , where $a$ and $n$ are constants. The graph of $\ln y$ against $\ln x$ , is a straight line passing through the points $(0.31, 4.02)$ and $(1.83, 3.22)$ , as shown in the diagram.

Taking natural logarithms on both sides for $2$

$\Rightarrow \ln (y) = \ln (a \times x^{n})$

Using the product rule of logarithms, $\ln (u v) = \ln u + \ln v$

$\Rightarrow \ln (y) = \ln (a) + \ln (x^{n})$

Using the power law of logarithms, $\ln {(u)}^{m} = m \ln (u)$

$a$

$\Rightarrow \ln (y) = n \ln (x) + \ln (a)$

On comparing the equation $\ln (y) = n \ln (x) + \ln (a)$ with $Y = m X + c$

$∴ Y = \ln (y), m = n, X = \ln (x), c = \ln (a)$

From the given points, gradient $m = n = (\frac{3.22 - 4.02}{1.83 - 0.31})$

$\Rightarrow n = - 0.5263$

$\Rightarrow n = - 0.53$ (rounded to the $2$ decimal places)

Now using one of the two points $(1.83, 3.22)$ , $m = - 0.53$ and $Y = m X + c$

$\Rightarrow 3.22 = - 0.53 \times 1.83 + c$

$\Rightarrow c = 3.22 + 0.9699$

$\Rightarrow c = 4.1899$

Therefore $\ln (a) = 4.1899$

Logarithms with base $e$ are called natural logarithms and if $y = e^{x}$ then $x = \ln y$

$\Rightarrow a = e^{4.1899}$

$\Rightarrow a = 66.0161$

$\Rightarrow a = 66.02$ (rounded to the $2$ decimal places)

Hence, $a = 66.02$ and $n = - 0.53$ ( $2$ decimal places.)

$t$	$10$	$20$	$30$	$40$	$y = e^{x}$
$m$	$40.9$	$33.5$	$27.4$	$22.5$	$18.4$

$t$	$2$	$4$	$6$	$8$	$10$
$T$	$63.3$	$57.7$	$52.8$	$48.7$	$45.2$

Important Questions on Logarithmic and Exponential Functions

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The variables x and y satisfy the equation y=a×xn, where a and n are constants. The graph of lny against lnx is a straight line passing through the points (0.31, 4.02) and (1.83, 3.22) as shown in the diagram. Find the value of a and the value of n correct to 2 significant figures.

Important Questions on Logarithmic and Exponential Functions

Learn and Prepare for any exam you want !