What is the inverse function of $f (x) = 10^{x}$ $f(x) =10^x$ ?

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Answer 1 · 2015-10-23T15:27:02

$y = {log}_{10} (x)$ $y =log_(10)(x)$

Let $y = 10^{x}$ $" " y = 10^x$

Take logs of both sides giving:

$ln (y) = ln (10^{x})$ $ln(y) =ln(10^x)$

Or better still use log to base 10 giving:

${log}_{10} (y) = {log}_{10} (10^{x})$ $log_(10)(y) =log_(10)(10^x)$

But ${log}_{10} (10^{x}) = x \times {log}_{10} (10)$ $log_(10)(10^x) = x times log_(10)(10)$

and ${log}_{10} (10) = 1$ $log_(10)(10) = 1$

Giving:

$log (y) = x$ $log(y) = x$

Now swap $x$ $x$ and $y$ $y$ around giving

$y = {log}_{10} (x)$ $y =log_(10)(x)$

What is the inverse function of f(x) =10^xf(x)=10xf(x) =10^x?