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Lecture 4

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MTH 162 Lecture 4: 5.4 General Logarithmic and Exponential Functions Notes
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University of Miami

Mathematics

MTH 162

Pachero

Fall

Description

MTH 162
Calculus II
5.4 General Logarithmic and Exponential Functions Notes
L. Sterling
Abstract
Provide a generalization to the terms listed in this section.
Solving Logarithmic and Exponential Theorems
Theorems
▯ log Theorem
a
– If both x > 0 and a > 0 while a 6= 1, then
y = f(x) = log x
a
and this only happens iﬀ if and only if that
y
x = a
▯ log is 1 ▯ to ▯ 1 Theorem
a
– If you let M, N, and even a being all positive real numbers while having a 6= 1, then
M = N
, and this only happens iﬀ if and only if that
loga(M) = log aN)
x
▯ a is 1 ▯ to ▯ 1 Theorem
– If you let both u and v be real numbers while having both a > 0 and a 6= 1, then
u = v
and this only happens iﬀ if and only if that
u v
a = a
Solving Logarithmic Equation Example
log 4 = 2log x
5 5
2log (x) = log (4)
2 log (x) log (4)
25 = 2
log (x) = log (2)
5 5
x = 2
1 Solving Exponential Equation Example
x
3 ▯ 3 = 240
3 ▯ 3 + 3 = 240 + 3
x
3 = 243
3 = 3 5
x = 5
Solving Logarithmic Equation Example
▯1 + log (x + 1) + log (x + 2) = 0
6 6
▯1 + log 6(x + 1)(x + 2)) = 0
▯1 + log ((x + 1)(x + 2)) + 1 = 0 + 1
6
log6((x + 1)(x + 2)) = 1
log6((x + 1)(x + 2)) = log6(6)
(x + 1)(x + 2) = 6
3
x + 2x + x + 2 = 6
x + 3x + 2 = 6
2
x + 3x + 2 ▯ 6 = 6 ▯ 6
x + 3x ▯ 4 = 0
(x ▯ 1)(x + 4) = 0
x = ▯4; 1
x = 1
Solving Exponential Equation Example
x x+3
5 = 4 ▯ ▯
ln(5 ) = ln 4 x+3

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