We want to evaluate the given log expressions
without using a calculator.
To do this, we'll form equations
by setting the log expressions equal to X.
Then we'll write the log equations
as the exponential equations
in order to determine the value of X.
So, for the first expression, we'll have log base five
of 0.008 equals X.
Again, if we can determine the value of X,
we can determine the value of the log expression.
Our next step is writhe the log equation
as an exponential equation
using the definition of a logarithm given here.
So for a quick review,
this log equation, this exponential equation,
are equivalent, or B is the base,
A is the exponent,
and N is the number.
So we're in the log equation as an exponential equation.
The base is five.
A logarithm is an exponent,
so we have five raise to the power of X
equals the number 0.008.
Another way to form the equation would be
to start with the base and the log equation
and work around the equal sign.
Five raised to the power of X equals 0.008.
Now we want to write both sides of the equation
with the same base.
So because five is prime,
let's try to write the right side with base five.
This decimal is eight thousandths.
Let's write the decimal as a fraction.
So we have five raised to the power of X
equals 8/1,000.
And now to simplify the fraction,
eight and 1,000 share a common factor of eight.
So let's write this as five of the X equals,
let's write eight as eight times one,
and 1,000 as eight times 125,
so we can see the greatest common factor.
Eight over eight simplifies to one.
So now we have five raised to the power of X
equals 1/125.
Let's look at the prime factorization in 125.
125 is equal to five times 25.
25 equals five times five.
So 125 is equal to five to the third.
So now we have five to the X
equals one over five to the third.
So they're both five of the third as the numerator.
It'll change the sign of the exponent.
And we'd have five to the X
equals five raised to the power of negative three.
And now because these are equal and the bases are the same,
the exponents must be equal,
and now we know X equals negative three.
So if X equals negative three,
our log expression is equals to negative three.
So this is equal to negative three.
For our homework system, do not include the equal sign.
Next we have log base 10 of 0.001.
So we'll set the expression equal to X.
By determining the value of X,
we can determine the value of the log expression.
So let's write the log equation as an exponential equation,
which would be 10 raised to the power of X
equals 0.001.
Again, 10 raised to the power of X equals 0.001.
Now I will write both sides of the equation
with the same base.
Notice how the decimal is one thousandth.
Let's write that as a fraction.
So 10 to the X equals 1/1,000.
1/1,000 is 10 to the third,
so we can write this as 10 to the X
equals one over 10 to the third.
One over to the third is the same as 10
raised to the power of negative three.
So we have 10 to the X
equals 10 to the power of negative three.
These are equal.
The bases are the same,
and therefore the exponents must be equal.
So X equals negative three.
Again, do not enter the equal sign in the online homework.
Next, we have log base 625 equals the square root of five.
So we'll form a log equation.
Log base of 625 of square root of five equals X.
Now we'll write the log equation as an exponential equation,
which would be 645 raised to the power of X
equals the square root of five,
plus write this using a rational exponent.
Because the exponent of five is one,
and the index for the square root is two,
we can write this as five raised to the power of 1/2.
So because five is prime,
let's work on running the left side as base five.
Well, 625 is equal to 25 times 25.
Of course 25 is equals to five times five.
So the prime factorization of 625 is five to the fourth.
So we can write the left side as five to the fourth
raised to the power of X equals five to the 1/2.
Now applying the power to power property of exponents,
here we multiply the exponents,
so we have five raised to the power of four times X,
which is just four X,
equals five raised to the power of 1/2.
Because these are equal and the bases are the same,
the exponents must be equal,
and therefore four X must equal 1/2.
So let's solve for X here.
Instead of dividing by four, let's multiply both sides
by the reciprocal of four, which would be 1/4.
So multiply the left side by 1/4
and the right side by 1/4.
Simplifying,
we just get X equals 1/8.
So here, the log expression is equal to 1/8.
Meaning, 625 to the 1/8 power
is equal to the square root of five.
For our last example,
our equation would be log base four of 32
equals X.
Again, by determining the value of X,
we can determine the value of the expression.
Then we'll write the log equation
as an exponential equation,
which should be the base of four
raised to the power of X equals 32.
Again, four raised to the power of X equals 32.
Now we're going to write both sides of the equation
with the same base,
because we cannot write 32 with the base of four.
Let's write four as two squared.
So we have two squared raised to the power of X equals,
let's see if we can write 32 with base two.
32 is to the fifth.
But if we don't know that,
we can always find the prime factors of 32.
32 is equal to 16 times two.
Two is prime.
16 equals four times four,
and four equals two times two.
I notice that 32 is two to the fifth.
I guess we have powers to powers on the left.
We multiply the exponents,
so the left side is two raised to the power of two times X,
just two X, equals two to the fifth.
These expressions are equal,
the bases are the same,
and therefore the exponents must be equal,
and therefore two X equals five.
Dividing both sides by two,
we have X equals 5/2.
So, a log base four of 32 equals 5/2.
Just remember to not enter the equal signs
into the answer cell.
I hope you found this helpful.