SOLVING LOGARITHM

Example 1

Solve the equation (1/2)^{2x + 1} = 1

Rewrite equation as (1/2)^{2x + 1} = (1/2)⁰

Leads to 2x + 1 = 0
Solve for x: x = -1/2

Example 2
Solve x y^m = y x³ for m

Divide all terms by x y and rewrite equation as: y^{m - 1} = x²

Take ln of both sides (m - 1) ln y = 2 ln x
Solve for m: m = 1 + 2 ln(x) / ln(y)

Example 3
Given: log₈(5) = b. Express log₄(10) in terms of b

Use log rule of product: log₄(10) = log₄(2) + log₄(5)
log₄(2) = log₄(4^1/2) = 1/2
Use change of base formula to write: log₄(5) = log₈(5) / log₈(4) = b / (2/3) , since log₈(4) = 2/3
log₄(10) = log₄(2) + log₄(5) = (1 + 3b) / 2

Example 4

Simplify without calculator: log₆(216) + [ log(42) - log(6) ] / log(49)

log₆(216) + [ log(42) - log(6) ] / log(49)
= log₆(6³) + log(42/6) / log(7²)
= 3 + log(7) /2 log(7) = 3 + 1/2 = 7/2

Example 5

Simplify without calculator: ((3^-1 - 9^-1) / 6)^1/3

((3^-1 - 9^-1) / 6)^1/3
= ((1/3 - 1/9) / 6)^1/3
= ((6 / 27) / 6)^1/3 = 1/3

Example 6

Express (log_xa)(log_ab) as a single logarithm

Use change of base formula: (log_xa)(log_ab)
= log_xa (log_xb / log_xa) = log_xb

Example 7

Find a so that the graph of y = log_ax passes through the point (e , 2)

2 = log_ae
a² = e
ln(a²) = ln e
2 ln a = 1
a = e^1/2

Reference:

http://www.analyzemath.com/high_school_math/grade_12/log_exp.html#solutions