Chapter 4.4, Problem 44E

Solution

To calculate: Find the inverse of the function.

  $y={10}^{x-6}$

Given information:

The logarithm $y=6+\log x$

Concept used:

1. Use the definition of logarithm ${\log }_{b}y=x$ if and only if $b^x=y$
2. Use the natural log that is written as $\ln$ instead of $lo{g}_e$ .

Calculation:

Observe that this function follow the condition the inverse property of logarithms that states,

  ${\log }_{b}y=x$ if and only if $b^x=y$

Therefore,

  $y=6+\log x$

Now switch $x$ and $y$ to find the inverse of the function and when a logarithm has no base , a base of $10$ assumed that is called common logarithm.

  $x=6+{\log }_{10}y$

Now use the condition

  ${10}^{x-6}=y$ or $y={10}^{x-6}$

Conclusion:

Hence, the solution is $y={10}^{x-6}$.