We want to evaluate the given log expressions.
Notice how for each log the base is not given,
so we know these are common logs,
meaning each log is base 10.
To help us evaluate these log expressions,
we'll form a log equation
by setting the expression equal to x,
then we'll write the log equation as an exponential equation
in order to solve for x.
For our first expression we have common log or log base 10
of 1/10000.
And, again, we'll set this equal to x.
If we can determine the value of x,
we'll know the value of the expression.
We'll apply the definition of a logarithm now
to write the log equation as an exponential equation.
Looking at our notes below,
these two equations are equivalent
where b is the base,
a is the exponent,
and n is the number.
So notice how the base is 10,
so we'd have 10,
a logarithm is an exponent,
so the exponent is x,
so we have 10 to the x equals the number
1/10000.
One way to verify the exponential equation
is to start with the base and work around the equal sign
to form the exponential equation.
Notice, 10 raised to the power of x
equals 1/10000.
So our exponential equation is correct.
In this form, we can solve for x
by writing both sides of the equation with the same base.
So let's work on writing the right side of the equation
with the base of 10.
So we have 10 to the x equals one over,
well, 10,000 is 10 to the fourth,
so we can write the right side as one over 10 to the fourth.
And now I can use a negative exponent
to rewrite the right side.
If we move 10 to the fourth up to the numerator,
it's going to change the sign of the exponent,
so we'd have 10 to the x
equals 10 to the power of negative four.
Because the expressions are equal
and the bases are the same,
the exponents must be equal,
and now we know x equals negative four.
So because x equals negative four,
we know the log expression equals negative four.
10 to the power of negative four equals 1/10000.
Next, we have common log or log base 10
of the cube root
of 10.
Again, let's set this equal to x to form a log equation
which we'll then write as an exponential equation.
Let's write the cube root of 10 using a rational exponent.
Because this is the cube root of 10 to the first,
we can write this as log base 10
of 10 to the 1/3 equals x.
Now we'll write the log equation as an exponential equation
where the base is 10,
the exponent is x,
and it's equal to the number 10 to the 1/3.
Again, we have 10 raised to the power of x
equals 10 to the 1/3.
And here we already have
both sides of the equation with base 10.
So because these are equal and the bases are the same,
we now know x equals 1/3.
And, again, this should make sense.
10 to the 1/3
equals the cube root of 10.
For our last expression,
we'll form an equation where we have
the common log or log base 10 of 0.001
equals x.
Now, we'll write the log equation as an exponential equation
where the base is 10, the exponent is x,
and this is going to equal the number 0.001.
But 0.001 would be 1/1000.
So let's write this as 10 to the x equals 1/1000.
Now, we'll write the right side with base 10.
So we'll have 10 to the x equals one over 10 to the third.
If we can write as 10 raised to the power of negative three
by moving 10 to the third up to the numerator.
So we'd have 10 to the x equal 10 to the negative three.
Because these are equal and the bases are the same,
the exponents must be equal,
and, therefore, x equals negative three.
Because these are common logs,
we can also easily check these on the graphing calculator.
So let's go ahead and do that before we go.
First, we have the common log of 1/10000,
which is negative four.
Next, we have the common log of the cube root of 10.
For cube root we press Math,
and then option four,
10, right arrow,
close parenthesis, Enter.
And to covert the decimal to a fraction,
we press Math, Enter, Enter.
1/3 is correct.
And then finally, we have the common log of 0.001,
which is negative three.
I hope you found this helpful.