Pre-Calculus - Week 5

Logarithmic Functions

Logarithms are the inverses of exponential functions. Every equation you could not solve last week by matching bases can be solved this week with logarithms. Beyond equation-solving, logarithms compress enormous scales into manageable numbers — earthquakes, sound intensity, pH — and they are the key to differentiating exponential functions in calculus.

Learning Objectives

- Convert between exponential and logarithmic form. - Evaluate common logarithms (base 10) and natural logarithms (base e). - Apply the three logarithm properties: product, quotient, and power rules. - Expand and condense logarithmic expressions. - Graph logarithmic functions and identify key features. - Solve logarithmic equations and check for extraneous solutions. - Apply the change-of-base formula. - Use logarithms to solve real-world exponential models.

1. Logarithm Definition

log_a(x) = y means a^y = x (read: log base a of x equals y). In other words: the logarithm is the exponent you put on the base a to get x. Example 1: log_2(8) = 3 because 2^3 = 8. Example 2: log_3(81) = 4 because 3^4 = 81. Example 3: log_5(1/25) = -2 because 5^(-2) = 1/25. Two special values: log_a(1) = 0 for any base (because a^0 = 1). log_a(a) = 1 for any base (because a^1 = a).

2. Common Log and Natural Log

Common logarithm: log(x) means log_10(x). Base 10 is the default when no base is written. Natural logarithm: ln(x) means log_e(x). Used everywhere in calculus. Key values: log(1) = 0, log(10) = 1, log(100) = 2, log(1000) = 3. ln(1) = 0, ln(e) = 1, ln(e^2) = 2, ln(e^(-1)) = -1. Inverse relationships: 10^(log(x)) = x and log(10^x) = x. e^(ln(x)) = x and ln(e^x) = x. These inverse cancellations are among the most used identities in calculus.

3. Converting Between Forms

Exponential form: a^y = x. Logarithmic form: log_a(x) = y. Example 1: Write 3^4 = 81 in log form. log_3(81) = 4. Example 2: Write log_2(32) = 5 in exponential form. 2^5 = 32. Example 3: Write e^2 = x in log form. ln(x) = 2. Practice converting freely in both directions — this is the core skill for solving equations.

4. Properties of Logarithms

Let a > 0, a is not 1, and x, y > 0. Product rule: log_a(xy) = log_a(x) + log_a(y). Quotient rule: log_a(x/y) = log_a(x) - log_a(y). Power rule: log_a(x^n) = n * log_a(x). Mnemonic: multiplication becomes addition, division becomes subtraction, exponents come out front. Warning: log_a(x + y) is NOT log_a(x) + log_a(y). There is no property for sums inside a log.

5. Expanding Logarithmic Expressions

Use the properties to write a single log as a sum/difference. Example 1: Expand log(x^3 * y / z^2). = log(x^3) + log(y) - log(z^2) = 3*log(x) + log(y) - 2*log(z) Example 2: Expand ln(sqrt(x) / (y^2 * z)). sqrt(x) = x^(1/2). = ln(x^(1/2)) - ln(y^2) - ln(z) = (1/2)*ln(x) - 2*ln(y) - ln(z) Expanding is useful for calculus: it lets you differentiate complex log expressions term by term.

6. Condensing Logarithmic Expressions

Reverse the properties to write a sum/difference as a single log. Example 1: Condense 3*log(x) - 2*log(y) + log(5). = log(x^3) - log(y^2) + log(5) = log(5x^3 / y^2) Example 2: Condense (1/2)*ln(x+1) - ln(3). = ln((x+1)^(1/2)) - ln(3) = ln(sqrt(x+1) / 3) Condensing is necessary before solving logarithmic equations.

7. Change-of-Base Formula

Calculators only have log (base 10) and ln (base e). To compute log_a(x): log_a(x) = log(x)/log(a) = ln(x)/ln(a) Example 1: Compute log_5(17). log_5(17) = log(17)/log(5) ≈ 1.2304/0.6990 ≈ 1.760. Example 2: Compute log_3(50). log_3(50) = ln(50)/ln(3) ≈ 3.912/1.099 ≈ 3.561. Verify: 3^3.561 ≈ 50. Check.

8. Graphing Logarithmic Functions

f(x) = log_a(x) for a > 1: - Domain: (0, inf). Logarithms are not defined for x <= 0. - Range: all reals. - x-intercept: (1, 0) since log_a(1) = 0. - Vertical asymptote: x = 0 (the y-axis). - Passes through (a, 1) and (1/a, -1). - Increasing if a > 1, decreasing if 0 < a < 1. f(x) = ln(x) passes through (1, 0), (e, 1), and approaches x = 0 as a vertical asymptote on the left.

9. Solving Logarithmic Equations

Method 1: One log on each side with the same base — set arguments equal. log_3(2x+1) = log_3(5). So 2x+1=5. x=2. Method 2: Isolate the log, then convert to exponential form. log_2(x - 1) = 4. Convert: 2^4 = x - 1. 16 = x - 1. x = 17. Method 3: Condense first, then convert. log(x) + log(x-3) = 1. log(x(x-3)) = 1. x(x-3) = 10^1 = 10. x^2 - 3x - 10 = 0. (x-5)(x+2)=0. x=5 or x=-2. Check: x=-2 gives log(-2) — undefined! Extraneous. Answer: x=5.

10. Applying Logs to Solve Exponential Equations

When bases cannot be matched, take ln of both sides. Example 1: Solve 3^x = 20. ln(3^x) = ln(20). x*ln(3) = ln(20). x = ln(20)/ln(3) ≈ 2.727. Example 2: Solve 5 * e^(2x) = 100. Divide by 5: e^(2x) = 20. ln both sides: 2x = ln(20). x = ln(20)/2 ≈ 1.498. Example 3: Solve 100 = 50 * e^(-0.1t) (half-life model). Divide by 50: e^(-0.1t) = 2. Take ln: -0.1t = ln(2). t = -ln(2)/0.1 = -6.931... Hmm, t negative means we want to know when 50g decays to 100g — that's impossible. Instead solve: 50 * e^(-0.1t) = 25. e^(-0.1t)=0.5. -0.1t = ln(0.5) = -0.693. t = 6.93 years.

11. Common Mistakes

- log(x + y) is NOT log(x) + log(y). There is no log property for addition inside. - log(x * y) is NOT log(x) * log(y). - Forgetting to check for extraneous solutions: x must be positive in log(x). - Confusing ln(x^2) = 2*ln(x) [valid for x>0] with ln(x^2) = 2*ln|x| when x could be negative. - Not applying log to the entire right side when solving: taking log of one side and not the other.