8.4 logarithmic functions
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Logarithms were invented to solve exponential equations like 2x = 6, where x is between 2 and 3. A logarithm with base b is defined as logby = x if and only if bx = y. Common properties of logarithms include logb1 = 0 and logbb = 1. Logarithmic functions have inverse relationships with exponential functions and can be graphed by shifting and reflecting the standard logarithmic curve.
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- If y = ax, then x = loga y, where loga y is the logarithm of y in base a.
- Rules for logarithms include: loga(xy) = loga x + loga y and loga(xn) = n loga x.
- Logarithms in base 10 are called common logarithms and are often written as log x, assuming base 10. Calculators have a log key for base 10 logarithms.
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8.4 logarithmic functions
- 2. What is a Logarithm? We know 22 = 4 and 23 = 8, but for what value of x does 2x = 6? It must be between 2 and 3… Logarithms were invented to solve exponential equations like this. x = log26 ≈ 2.585
- 3. Logarithms with Base b Let b and y be positive numbers and b≠1. The logarithm of y with base b is written logby and is defined: logby = x if and only if bx = y
- 4. Rewriting Log Equations Write in exponential form: log2 32 = 5 log5 1 = 0 log10 10 = 1 log10 0.1 = -1 log1/2 2 = -1
- 5. Special Log Values For positive b such that b ≠ 1: Logarithm of 1: logb 1 = 0 since b0 = 1 Logarithm of base b: logb b = 1 since b1 = b
- 6. Evaluating Log Expressions To find logb y, think “what power of b will give me y?” Examples: log3 81 log1/2 8 log9 3
- 7. Your Turn! Evaluate each expression: log4 64 log32 2
- 8. Common and Natural Logs Common Logarithm - the log with base 10 Written “log10” or just “log” log10 x = log x Natural Logarithm – the log with base e Can write “loge“ but we usually use “ln” loge x = ln x
- 9. Evaluating Common and Natural Logs Use “LOG” or “LN” key on calculator. Evaluate. Round to 3 decimal places. log 5 ln 0.1
- 10. Evaluating Log Functions The slope s of a beach is related to the average diameter d (in mm) of the sand particles on the beach by this equation: s = 0.159 + 0.118 log d Find the slope of a beach if the average diameter of the sand particles is 0.25 mm.
- 11. Inverses The logarithmic function g(x) = logb x is the inverse of the exponential function f(x) = bx. Therefore: g(f(x)) = logb bx = x and f(g(x)) = blogb x = x This means they “undo” each other.
- 12. Using Inverse Properties Simplify: 10logx log4 4x 9log9 x log3 9x
- 13. Your Turn! Simplify: log5 125x 5log5 x
- 14. Finding Inverses Switch x and y, then solve for y. Remember: to “chop off a log” use the “circle cycle”! Find the inverse: y = log3 x y = ln(x + 1)
- 15. Your Turn! Find the inverse. y = log8 x y = ln(x – 3)
- 16. Logarithmic Graphs Remember f-1 is a reflection of f over the line y = x. Logs and exponentials are inverses! exp. growth exp. decay
- 17. Properties of Log Graphs General form: y = logb (x – h) + k Vertical asymptote at x = h. (x = 0 for parent graph) Domain: x > h Range: All real #s If b > 1, graph moves up to the right If 0 < b < 1, graph moves down to the right.
- 18. To graph: Sketch parent graph (if needed). Always goes through (1, 0) and (b, 1) Choose one more point if needed. Don’t cross the y-axis! Shift using h and k. Be Careful: h is in () with the x, k is not
- 19. Examples: Graph. State the domain and range. y = log1/3 x – 1 Domain: Range:
- 20. Graph. State the domain and range. y = log5 (x + 2) Domain: Range:
- 21. Your Turn! Graph. State the domain and range. y = log3 (x + 1) Domain: Range: