Solution of Exponential Equations Using Logarithms (VCE SSCE Mathematical Methods): Flashcards

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Solution of Exponential Equations Using Logarithms
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Equivalence of ax=ba^x = b in logarithmic form

logab=x\log_a b = x

Why logarithms solve exponential equations

They extract values from exponents

Convert 2x=112^x = 11 to logarithmic form

x=log211x = \log_2 11

Inequality direction when base a>1a > 1 in axba^x \geq b

Stays the same

Inequality direction when 0<a<10 < a < 1 in axba^x \geq b

Reverses

Why inequality reverses when 0<a<10 < a < 1

Function is decreasing

Type of function when base a>1a > 1

Strictly increasing

Type of function when 0<a<10 < a < 1

Strictly decreasing

Conditions for logab\log_a b to exist

a>0a > 0, a1a \neq 1, and b>0b > 0

First step for complex eqns like 32x1=283^{2x-1} = 28

Isolate exponential, then convert to log form

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