Question

What is the inverse of y= 10^x +8?

Answer 1

Answer:

$x = \log_{10}(y - 8)$, where $y > 8$.

Step-by-step explanation:

The question is asking for an expression for $x$ in terms of $y$.

The first step is to move as many constants away from the side with $x$ as possible. In this question, that could be done by subtracting $8$ from both sides of the equation.

$y - 8 = 10^{x} + 8 - 8$.

$y - 8 = 10^{x}$.

Notice how $x$ is in the position of an exponent. Logarithms could help move $x$ out of that position.

For any base $b > 0$:

$\log_{b} b^{x} = x\, \log_{b}\, b = x$.

In this question, $10$ would be the base. That is: $b = 10$.

Take the logarithm (with $10$ as the base) of both sides of the equation $y - 8 = 10^{x}$:

• Left-hand side: $\log_{10} (y - 8)$.
• Right-hand side: $\log_{10} \left(10^{x}\right) = x\, \log_{10}\left(10\right) = x$.

Equate both sides to find an expression for $x$:

$x = \log_{10}(y - 8)$.