The question is:
Are these two log expressions equal to each other?
And the answer is no, they are not.
Let's first verify this using a calculator
and then we'll come back
and evaluate both of these logarithms by hand.
The expression on the left is log base 2 of 4 cubed,
which is equal to 6.
The expression on the right is the cube of log base 2 of 4,
which is equal to 8.
Well, of course, 6 doesn't equal 8,
and therefore the two log expressions are not equal.
And that's how you evaluate both logarithms by hand.
Let's begin with log base 2 of 4 cubed.
Notice,
the exponent of 3 is inside the parentheses on the 4.
This is the form where we can apply the power property
of logarithms shown here below:
log base b of x to the power of y is equal
to y times log base b of x,
which means log base 2 of 4 cubed is equal
to 3 times log base 2 of 4.
And asked of you how
to we evaluate log base 2 of 4 on the side.
To do this we'll form a log equation
where we say log base 2 of 4 equal to x.
If we determine the value of x,
we know the value of the logarithm.
To solve this log equation,
we'll write the corresponding exponential equation
using the definition of a logarithm shown here below,
where y equals log base b of x is equivalent
to b to the power of y equals x.
So in our case, 2 is the base,
x is the exponent, and it's equal to 4.
So the corresponding exponential equation
is 2 to the power of x equals 4
and 4 is equal to 2 squared.
So because these are equal and the bases are the same,
we know x must equal 2,
which means log base 2 of 4 is equal to 2.
So going back to our expression,
3 times log base 2 of 4 is equal to 3 times 2, which is 6.
Another way to evaluate this logarithm
would have been to just cube 4.
4 cubed is equal to 64.
So we could have written this as log base 2 of 64.
And then to evaluate this logarithm,
we can use the same approach we did for log base 2 of 4.
Let's do this down here.
Log base 2 of 64 is equal to x.
The corresponding exponential equation
is 2 to the power of x equals 64
and 64 is equal to 2 to the 6th power.
2 to the x is equal to 2 to the 6th,
giving us x is equal to 6.
So a log base 2 of 64 is equal to 6,
which, of course, is what we expect.
And now let's evaluate the logarithm on the right,
which is the cube of log base 2 of 4.
So the exponent of 3 here is not attached to the 4.
It's actually attached to the logarithm.
We're cubing log base 2 of 4,
meaning we have three factors of log base 2 of 4,
which can be written like this.
We can also attach the exponent of 3 on the logarithm
and write the expression in this form here.
This is the cube of log base 2 of 4.
But to evaluate this by hand,
I think this form here is easiest to use.
So the given expression, the cube of log base 2 of 4
is equal to, well, we know log base 2 of 4 is 2.
So this just gives us 2 cubed, which is equal to 8.
So it is important to recognize the difference
between these two expressions.
Before we go, let's summarize what we discovered.
When evaluating a logarithm where the exponent of y
is inside the parentheses attached to x in this form here,
we can apply the power property of logarithms
and write the log as y times log base b of x.
But if the exponent of y is outside the parentheses,
it actually applies to the entire logarithm
that can be written in either of the two forms shown here
on the right.
I hope you found this helpful.