Logarithm Problem

Question

Using the facts  log_b 11=1.232  and log_b  4 =0.712 evaluate each of the following WITHOUT using a calculator.

Use the decimals provided.  Do not round any answers.

log_b 44 = Answer

log_b (11/4) = Answer

log_b (11^4) = Answer

Answer 1

Logarithm identities are needed to solve this problem. After solving, I began explaining what a logarithm is, but I ran out of time!

Basically, because the logarithm represents the exponent needed to connect two numbers, it has operations similar to an exponent. In fact, the logarithm is the opposite of the exponent.

If two logarithms (with the same base!) are added, this is similar to adding exponents. Adding exponents happens when you multiply two things together! This is why we can turn the logarithm of a product into a sum of logarithms.

If a logarithm is multiplied by a number, this is like multiplying an exponent by a number! We multiply exponents when we raise our expression to a second exponent. This is why we can turn the logarithm of a number raised to a power into the power times that logarithm.

Here are some examples of the exponential operations and their logarithmic equivalents:

5^2 * 5^3 = 5^(2+3)

log_5(2 * 3) = log_5(2) + log_5(3)

(5^2)^3 = 5^(2*3)

log_5(2^3) = 3*log_5(2)

See the relationship?