Properties of logs. Why does the log of a power, does the exponent become a coefficient? I'm having trouble seeing number 3 clearly. Or rather, I'm having trouble seeing a proof. I've tested a series of numbers and it works, but I'm having trouble seeing why. Can anyone clarify?



Here is where I get confused:

I start with this log:

$$ \log_2 4^1 = 2$$

now if I raise both sides to the 2nd power then:

$$ \log_2 4^{1^2} = 2^2$$
$$ = \log_2 4^2 = 4$$
$$ = 2 \cdot \log_2 4 = 4$$

but if I raise both sides to the power of 1.5 then:
$$ \log_2 4^{2 \cdot 1.5} \ne 4^{1.5} \ne 8$$
By definition:
$$
a^{\log_a x}=x
$$

Now, if:
$$
\log_a (x^y)=\alpha
$$
we have
$$
a^\alpha=x^y =\left( a^{\log_a x}\right)^y = a^{y\log_a x}
$$
so
$$
\alpha=y\log_a x
$$
Logy=logx^a

e^y=x^a

e^(y/a)=x

y/a=logx

y=alogx

You have raised the power in wrong way for log one answer is same