Definition of a logarithm:
log a = x ——>b^x=a
b
So, by this definition, b=base, a=product, and x=exponent. Therefore, you can remember: a logarithm IS an exponent or log=EXP. This should make sense because a logarithm solves for an exponent given the base and product, which makes it the inverse operation of exponent. The definition at the top can be translated between logarithms and the equivalent exponential form.
Let’s solve the equation now using these concepts:
log (1/64) = x
4
Let’s convert into exponential form. We will do this to see the logarithm work as an exponential equation, and we are able to do this because they are equivalent forms.
log (1/64) = x ————> 4^x=1/64
4
So, we are actually looking for an exponent with a base of 4 that produces 1/64. Notice that 1/64 is less than 1, and the only way to obtain a product of less than 1 with exponents is to use negative exponents. Remember, positive exponents are how many times the base multiplies itself; this concept applies to negative exponents, but a negative exponent causes the base to be reciprocated.
We know that 4³=64, so how does 64 become 1/64? Well, an exponent of -3 caused this to occur. This is because 4^-3=1/4³=1/64. Remember, negative exponents must be moved to the denominator or numerator to make the power positive; every rational number can be made into a fraction by dividing by 1 because it doesn’t change the value of the number.
Thus, x must be -3. Let’s check:
x=-3
4^-3=1/64
1/4³=1/64
1/64=1/64
So, x=-3 is the answer.