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 $y$ satisfies the equation $y = K (b^{x})$ , where $K$ and $b$ are constants. The graph of $\ln y$ against $x$ is a straight line passing through the points $(2.3, 1.7)$ and $(3.1, 2.1)$ , as shown in the diagram. Find the values of $K$ and $b$ , correct to $2$ decimal places.

Accepted Answer

Given the variables $x$ and $y$ satisfy the equation $y = K (b^{x})$ , where $K$ and $b$ are constants. The graph of $\ln y$ against $x$ is a straight line passing through the points $(2.3, 1.7)$ and $(3.1, 2.1)$ , as shown in the diagram.

Taking natural logarithms on both side for $y = K b^{x}$

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

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

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

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

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

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

On comparing the above equation with $Y = m X + c$

$\Rightarrow Y = \ln (y), m = \ln (b), c = \ln (K)$

From the given points we get gradient $m = \ln (b) = (\frac{2.1 - 1.7}{3.1 - 2.3})$

$\Rightarrow m = \frac{0.4}{0.8}$

$\Rightarrow m = 0.5$

$∴ \ln (b) = 0.5$

$\Rightarrow b = 1.648$

$\Rightarrow b = 1.65$ (rounded to $2$ decimal places)

Now using one of the two points $(2.3, 1.7)$ , $m = 0.5$ and $Y = m X + c$

$\Rightarrow 1.7 = 0.5 \times 2.3 + c$

$\Rightarrow c = 0.55$

$∴ \ln (K) = 0.55$

$\Rightarrow K = 1.733$ (rounded to $2$ decimal points)

Hence $K = 1.73$ and $b = 1.65$

The variables $x$ and $y$ satisfies the equation $y = K (b^{x})$ , where $K$ and $b$ are constants. The graph of $\ln y$ against $x$ is a straight line passing through the points $(2.3, 1.7)$ and $(3.1, 2.1)$ , as shown in the diagram. Find the values of $K$ and $b$ , 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 satisfies the equation y=Kbx, where K and b are constants. The graph of ln y against x is a straight line passing through the points (2.3, 1.7) and (3.1, 2.1), as shown in the diagram. Find the values of K and b, correct to 2 decimal places.

Important Questions on Logarithmic and Exponential Functions

Learn and Prepare for any exam you want !

The variables $x$ and $y$ satisfies the equation $y = K (b^{x})$ , where $K$ and $b$ are constants. The graph of $\ln y$ against $x$ is a straight line passing through the points $(2.3, 1.7)$ and $(3.1, 2.1)$ , as shown in the diagram. Find the values of $K$ and $b$ , correct to $2$ decimal places.