Question

How to express exponential equation, 3^c = 27 , as a logarithmic equation

Answer 1

$log_{3}$27 = c

using the law of logs : $log_{b}$ x = n ⇔ x = $b^{n}$

given $3^{c}$ = 27 then

$log_{3}$ 27 = c

Answer 2

If you consider the logarithm base 3 of both sides, you have

$\log_3(3^c) = \log_3(27)$

You can use a rule of logarithms that allow you to turn exponents into multiplicative factors:

$\log_a(b^c) = c\log_a(b)$

So the equation becomes

$c\log_3(3)=\log_3(27)$

Now, by definition, you have

$\log_a(b)=c \iff a^c=b$

So, you have

$\log_3(3)=x \iff 3^x=3 \iff x = 1,\quad \log_3(27)=y \iff 3^y=27\iff y=3$

So, the equation becomes

$1\cdot c = 3 \iff c=3$