Algebra and Precalculus Help: Introduction to Logarithms

A math concept the advanced math student will be expected to master is logarithms. Depending upon the math curriculum, this concept is introduced either in algebra or precalculus. In this article I will explain what logarithms are and give examples of basic math problems the algebra or precalculus student will be expected to solve.

What is a logarithm?
"Logarithm" is just another name for exponent or power. If we look at the algebraic equation:

y = b^x

We say that "base b is raised to the power or exponent of x."

We can rewrite this algebraic equation as:

log_by = x

We say that the exponent x is also called "the logarithm of y to the base b."

To summarize more formally:

x = log_b y if and only if y = b^x

What is the relationship between two logarithms that have the same base?
This is a basic principle the algebra student should memorize:

log_bx₁ = log_bx₂ if and only if x₁ = x₂

Sample Basic Logarithm Math Problems
Here are some basic logarithm math problems in increasing difficulty that the algebra or precalculus student should know how to solve.

Express 2³ = 8 in logarithmic form.
Answer: log₂8 = 3

Express 10^-2 = 0.01 in logarithmic form.
Answer: log₁₀0.01 = -2

Express log₃81 = 4 in exponential form.
Answer: 3⁴ = 81

Express log₅(1/25) = -2 in exponential form.
Answer: 5^-2 = (1/25)

Find the logarithm for log₂16.
Answer: 2⁴ = 16, so the answer is 4.

Find the logarithm for log₃(1/27).
Answer: 3^-3 = (1/27), so the answer is -3.

Find the logarithm for log₈1.
Answer: 8⁰ = 1, so the answer is 0.

Solve for x. log_x64 = 3.
Answer:
Convert to exponential form first. x³ = 64.
Take the cube root of 64 to find the answer is 4.

Solve for x. log_x27 = (¾).
Answer:
Convert to exponential form first. x^(3/4) = 27.
Raise both sides to the (4/3) power. This gives x = 27^(4/3).
We know from our basic math facts that 3³ = 27, so we can rewrite our equation as x =3^3*(4/3) = 3⁴ = 81.

Solve for x. log_x3 = -2.
Answer:
Convert to exponential form first. x^-2 = 3.
Raise both sides to the (-1/2) power. This gives x = 3^(-1/2)

Solve for x. log_2x125 = 3.
Answer:
Convert to exponential form first. (2x)³ = 125
Simplify. 8x³ = 125
Simplify. x³ = (125/8)
Our basic math facts tells us that 5³ = 125 and 2³ = 8.
Take the cube root of both sides. x = (5/2)

Solve for x. log₃(2x + 1) = 4.
Answer:
Convert to exponential form first. 3⁴ = 2x + 1
Simplify. 81 = 2x + 1
Using basic algebra simplification we get x = 40.

Solve for x. log₂16 = 3x -1.
Answer:
Convert to exponential form first. 2^(3x-1) = 16.
We know from our basic math facts that 2⁴ = 16.
Since the logarithms (or exponents) are of the same base 2, we can say 3x-1 = 4.
Simplify. x = (5/3).

Blessings!

Source
Mary P. Dolciani, et. al. Algebra 2 and Trigonometry. Revised Edition