How do you find the inverse of #f(x) =10^x#?

Question

Sadie Alexander · Accepted Answer

#f^(-1)(x)=logx#

By definition, #y=f^(-1)(x)ifff(y)=x# #iff 10^y=x# We can now take log base 10 on both sides and use laws of logs to yield #y=log_10x#

I will also show the graphs of both f and its inverse for clarity.

graph{10^x [-6.37, 6.12, -2.67, 3.57]}

graph{logx [-2.26, 6.514, -2.257, 2.126]}

Sadie Ackerman · Answer

undefined To find the inverse of a function $ f(x) $, which is denoted as $ f^{-1}(x) $, follow these steps:

1. Start with the original function: $ f(x) = 10^x $.
2. Replace $ f(x) $ with $ y $: $ y = 10^x $.
3. Swap the roles of $ x $ and $ y $: $ x = 10^y $.
4. Solve the equation for $ y $ to find the inverse function $ f^{-1}(x) $.

To solve for $ y $, take the logarithm base 10 (or any other logarithm base) of both sides:

$$ \log_{10}(x) = \log_{10}(10^y) $$

By the logarithmic property $ \log_a(a^b) = b $, we have:

$$ \log_{10}(x) = y $$

Therefore, the inverse function of $ f(x) = 10^x $ is:

$$ f^{-1}(x) = \log_{10}(x) $$

Alternatively, you can write the inverse function as $ f^{-1}(x) = \log(x) $ if the base of the logarithm is not explicitly stated, as it is common in many mathematical contexts to assume base 10 logarithms unless otherwise specified.