Ex: Evaluate Logarithmic Expressions without a Calculator – Common Log

Ex: Evaluate Logarithmic Expressions without a Calculator – Common Log


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.

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