ALGEBRA 2 LECTURE E – 2:
Introduction to Logarithmic Functions
Reading Assignment: Chapter 6, Pages 370 – 388
LOGARITHMIC FUNCTIONS
Logarithms are used to find unknown exponents in exponential models.
Logarithmic functions define many measurement scales in the sciences, including the pH, decibel, and
Richter scales.
The Logarithmic Function y = logbx with base b, or x = by, is the inverse of the exponential
function y
= bx, where b  1 and b  0.
EQUIVALENT EXPONENTIAL AND LOGARITHMIC FORMS
For any positive base b, where b  1:
bx = y if and only if x = logby
EXAMPLE 1:
A. Write 53 = 125 in logarithmic form.
B. Write log381 = 4 in exponential form.
TRY THIS PAGE 371: Complete the table below:
EXPONENTIAL
FORM
LOGARITHMIC
FORM
3– 2 = 1/9
25 = 32
log101000 = 3
log164 = ½
EXAMPLE 2: Solve 10x = 85 for x. Round your answer to the nearest thousandth.
TRY THIS PAGE 371: Solve 10x = 1/109 for x. Round your answer to the nearest thousandth.
ALGEBRA 2 LECTURE E – 2:
Introduction to Logarithmic Functions
ONE-to-ONE PROPERTY OF EXPONENTS
If bx
= by, then x = y
EXAMPLE 3: Find the value of v in each equation.
A. v = log1255
B. 5 = logv 32
C. 4 = log3v
TRY THIS PAGE 373: Find the value of v in each equation.
A. v = log464
B. 2 = logv 25
C. 6 = log3v
PROPERTIES OF LOGARITHMS
PROPERTY
EXPONENTS
LOGARITHMS
For m  0, n  0, b  0, and b  1
PRODUCT PROPERTY
am  an = am+n
𝐥𝐨𝐠 𝒃 (𝒎𝒏) = 𝐥𝐨𝐠 𝒃 𝒎 + 𝐥𝐨𝐠 𝒃 𝒏
QUOTIENT PROPERTY
𝒂𝒎
= 𝒂𝒎−𝒏
𝒏
𝒂
POWER PROPERY
m n
(a ) = a
mn
𝐥𝐨𝐠 𝒃
𝒎
= 𝐥𝐨𝐠 𝒃 𝒎 − 𝐥𝐨𝐠 𝒃 𝒏
𝒏
For any real number p:
𝐥𝐨𝐠 𝒃 𝒎𝒑 = 𝒑 𝐥𝐨𝐠 𝒃 𝒎
EXAMPLE 4: Given log23  1.5850, approximate the value of each expression below using the
Product and Quotient Properties of logarithms.
A. log212
B. log21.5
TRY THIS PAGE 379: Given log23  1.5850, approximate the value of each expression below using
the Product and Quotient Properties of logarithms.
3
A. log218
B. log 2 4
ALGEBRA 2 LECTURE E – 2:
Introduction to Logarithmic Functions
EXAMPLE 5: Write each expression as a single logarithm. Then simplify, if possible.
A. log 3 10 − log 3 5
B. log 𝑏 𝑢 + log 𝑏 𝑣 − log 𝑏 𝑢𝑤
TRY THIS PAGE 379: Write each expression as a single logarithm. Then simplify, if possible.
A. log 4 18 − log 4 6
B. log 𝑏 4𝑥 + log 𝑏 𝑦 − log 𝑏 3𝑦
EXAMPLE 6: Evaluate log 5 254 using the power property of logarithms.
TRY THIS PAGE 380: : Evaluate log 3 27100 using the power property of logarithms.
EXPONENTIAL – LOGARITHMIC INVERSE PROPERTIES
For b  1 and b  0:
logbbx = x and blogbx = x for x  0
EXAMPLE 7: Evaluate each expression
A. 3log3 4 + log 5 25
B. log 2 32 − 5log5 3
TRY THIS PAGE 380: Evaluate each expression
A. 7log7 11 − log 3 81
B. log 8 85 + 3log3 8
ALGEBRA 2 LECTURE E – 2:
Introduction to Logarithmic Functions
ONE-to-ONE PROPERTY OF EXPONENTS
If logbx
= logby, then x = y
EXAMPLE 8: Solve log 3 (𝑥 2 + 7𝑥 − 5) = log 3 (6𝑥 + 1) for x.
TRY THIS: Solve log 3 𝑥 = log 3 (2𝑥 − 4) for x.
CHANGE OF BASE FORMULA
For any positive real numbers a  1, b  1, and x  0:
log 𝑏 𝑥 =
log𝑎 𝑥
log𝑎 𝑏
EXAMPLE 9: Evaluate log 7 56
TRY THIS PAGE 388: Evaluate log 8 36
HW E – 2
Page 374 #13, 21, 25, 31, 37, 39, 45, 49, 57, 69, 79;
Page 382 #17, 27, 35, 39, 43, 49, 55, 61;
Page 389 #11, 23, 29, 39