Question

To sketch the graph the functions y= log₁.₅ x, y= lnx , y = log₁₀ x and y= log₅₀ x on a common screen and interpret how these graphs are related.

Short answer

The graphs are related in such a way that \({\log _{{\rm{base }}}}x\) is \(y = \ln x\) times the factor.

That is, \(y = \frac{1}{{\ln ({\rm{ base }})}}\).

Step-by-step solution

Given data

The given functions are y= log₁.₅ x, y= lnx , y = log₁₀ x and y= log₅₀ x

Concept of Logarithm

Logarithm is the exponent or power to which a base must be raised to yield a given number.

For all a > 0and a ≠ 1 , log_a x = ln x / ln a

Plot of the graph of given function

The graph of the functions, y= log₁.₅ x, y= lnx , y = log₁₀ x and y= log₅₀ x are drawn and shown below in Figure as follows.

Calculation to identify the relationship between the each graph

From figure, it is noted that the graphs are passing through the common point \((1,0)\).

Use the formula, \({\log _b}a = \frac{{\ln a}}{{\ln b}},b \ne 1\) and express the given log function in terms of natural logarithm (In) and identify the relationship between each graph. \[\begin{aligned}{\log _{1.5}}x = \frac{{\ln x}}{{\ln 1.5}}\\{\log _{10}}x = \frac{{\ln x}}{{\ln 10}}\\{\log _{50}}x = \frac{{\ln x}}{{\ln 50}}\end{aligned}\]

Thus, the given graphs are related in such a way that \({\log _{{\rm{base }}}}x\) is \(y = \ln x\) times the factor.

That is, \(y = \frac{1}{{\ln ({\rm{ base }})}}\).