Logarithms and Exponentials

Grade 11 · Math · Study Guide

Master the relationship between exponential and logarithmic functions, including laws of logarithms and solving exponential equations.

Exponential and logarithmic functions are inverses of each other — a fact that unlocks powerful problem-solving techniques. An exponential function has the form f(x) = b^x, where b > 0 and b ≠ 1. The base b determines whether the function grows (b > 1) or decays (0 < b < 1). Common bases include 10 (common logarithm) and e ≈ 2.718 (natural logarithm, ln).

A logarithm answers the question: to what power must I raise the base to get this number? Formally, log_b(x) = y means b^y = x. This inverse relationship means you can rewrite any exponential equation as a logarithm and vice versa. For example, 2^3 = 8 becomes log_2(8) = 3.

The three Laws of Logarithms are essential tools: the Product Rule (log_b(MN) = log_b(M) + log_b(N)), the Quotient Rule (log_b(M/N) = log_b(M) - log_b(N)), and the Power Rule (log_b(M^n) = n times log_b(M)). The Change of Base Formula — log_b(x) = ln(x)/ln(b) — lets you evaluate any logarithm on a standard calculator.

Exponential equations appear constantly in science: radioactive decay, population growth, compound interest, and pH calculations all rely on this family of functions. When solving, isolate the exponential expression and then apply a logarithm to both sides.

Key terms

Logarithm
The inverse of an exponential: log_b(x) = y means b^y = x.
Natural Logarithm (ln)
A logarithm with base e (approx 2.718), written ln(x) = log_e(x).
Change of Base Formula
log_b(x) = log(x)/log(b) = ln(x)/ln(b); converts any log to a computable form.
Exponential Decay
A function of the form f(t) = A times b^t where 0 < b < 1, modeling quantities that decrease over time.
Asymptote
A line a curve approaches but never reaches; exponential functions have a horizontal asymptote at y = 0.
Inverse Functions
Two functions that undo each other; b^x and log_b(x) are inverses, so their graphs reflect across y = x.

Flashcards

log_b(x) = y
Logarithmic form: b raised to the power y equals x.
Product Rule
log_b(MN) = log_b(M) + log_b(N); multiplying inside equals adding logarithms.
Quotient Rule
log_b(M/N) = log_b(M) - log_b(N); dividing inside equals subtracting logarithms.
Power Rule
log_b(M^n) = n times log_b(M); an exponent inside the log moves to a coefficient outside.
log_b(b) = 1
Any logarithm with the same argument and base equals 1, because b^1 = b.
log_b(1) = 0
Any logarithm of 1 equals 0, because b^0 = 1 for any valid base b.
Compound Interest Formula
A = P(1 + r/n)^(nt); exponential growth models money growing over time.
e (Euler's Number)
The mathematical constant approximately 2.718; the base of natural exponential functions and ln.

Questions

  1. Which expression is equivalent to log base 2 of 32?

    1. 4
    2. 5
    3. 6
    4. 8
    Show answer

    Answer: 5

    2^5 = 32, so log base 2 of 32 equals 5.

  2. Using the Product Rule, log base 3 of 9 plus log base 3 of 3 equals log base 3 of ___.

    Show answer

    Answer: 27

    Product Rule: log_3(9) + log_3(3) = log_3(9 times 3) = log_3(27). Check: 3^3 = 27.

  3. Solve for x: 5^x = 125

    1. x = 2
    2. x = 3
    3. x = 4
    4. x = 5
    Show answer

    Answer: x = 3

    125 = 5^3, so 5^x = 5^3 means x = 3.

  4. The natural logarithm ln(1) equals 0.

    Show answer

    Answer: True

    ln(1) = log_e(1) = 0 because e^0 = 1 for any base.

  5. Using the Change of Base Formula, log base 4 of 64 equals:

    1. 2
    2. 3
    3. 4
    4. 16
    Show answer

    Answer: 3

    log_4(64) = ln(64)/ln(4) = 3 times ln(4) divided by ln(4) = 3. Verify: 4^3 = 64.

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