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 $y = K x^{m}$ , where $K$ and $m$ are constants. The graph of $\ln y$ against $\ln x$ is a straight line passing through the points $(0, 2.0)$ and $(6, 10.2)$ , as shown in the diagram. Find the values of $K$ and $m$ , correct to $2$ decimal places.

Accepted Answer

Given the variables $x$ and $y$ satisfy the equation $y = K x^{m}$ , where $K$ and $m$ are constants. The graph of $\ln y$ against $\ln x$ is a straight line passing through the points $(0, 2.0)$ and $(6, 10.2)$ , as shown in the diagram.

Taking natural logarithms on both sides for $y = K x^{m}$

$\Rightarrow \ln (y) = \ln (K x^{m})$

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

$\Rightarrow \ln (y) = \ln (K) + \ln (x^{m})$

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

$\Rightarrow \ln (y) = \ln (K) + m \ln (x)$

$\Rightarrow \ln (y) = m \ln (x) + \ln (K)$

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

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

From the given points we get gradient $m = (\frac{10.2 - 2.0}{6 - 0})$

$\Rightarrow m = 1.3666$

$\Rightarrow m = 1.37$ (rounded to $2$ decimal places)

Now using one of the two points $(0, 2.0)$ , $m = 1.37$ and $Y = m X + c$

$\Rightarrow 2 = 1.37 \times 0 + c$

$\Rightarrow c = 2$

Therefore $\ln (K) = 2$

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

$\Rightarrow K = e^{2}$

$\Rightarrow K = 7.3890$

$\Rightarrow K = 7.39$ (rounded to $2$ decimal places)

Hence, $K = 7.39$ and $m = 1.37$ to $2$ decimal places.

The variables $x$ and $y$ satisfy the equation $y = K x^{m}$ , where $K$ and $m$ are constants. The graph of $\ln y$ against $\ln x$ is a straight line passing through the points $(0, 2.0)$ and $(6, 10.2)$ , as shown in the diagram. Find the values of $K$ and $m$ , correct to $2$ decimal places.

Important Questions on Logarithmic and Exponential Functions

Learn and Prepare for any exam you want !

The variables x and y satisfy the equation y=Kxm, where K and m are constants. The graph of ln y against ln x is a straight line passing through the points (0, 2.0) and (6, 10.2), as shown in the diagram. Find the values of K and m, correct to 2 decimal places.

Important Questions on Logarithmic and Exponential Functions

Learn and Prepare for any exam you want !

The variables $x$ and $y$ satisfy the equation $y = K x^{m}$ , where $K$ and $m$ are constants. The graph of $\ln y$ against $\ln x$ is a straight line passing through the points $(0, 2.0)$ and $(6, 10.2)$ , as shown in the diagram. Find the values of $K$ and $m$ , correct to $2$ decimal places.