Unit 9.3 Functions of Logarithms and Solving Logarithmic Equations
Product Property
A logarithmic function that has two values multiplied together can be broken apart using addition
log (m · n) = log (m) + log (n)
ln (m · n) = ln (m) + ln (n)
This is not to be confused with log (m + n) which cannot be broken down further. It is also important to note that when recombining into a single log expression, they need to have the same base.
Product property of logarithms, solve for x
Solve for x : log (x) + log (2) = 2
Combine using the Product rule
log (x · 2) = 2
To cancel out the log, take it to the base power (rule 4) on both sides
10log(2x) = 102
The 10 and the log cancel out
10log(2x) = 102
2x = 100
x = 50
Quotient Rule
A logarithmic function that has two values divided can be broken apart using subtraction
[latex]\log{\left(\ \frac{m}{n}\ \right)=\log{\left(m\right)-\log(n)}}[/latex]
[latex]\ln{\left(\ \frac{m}{n}\ \right)=\ln{\left(m\right)-\ln(n)}}[/latex]
Logarithmic Equations with Different Bases
To solve an exponential equation with different bases, we can use natural log or common log and our rules to simplify
Examples
Solve 4-7x = 11x + 3 for x
Take the ln of both sides
ln (4-7x) = ln (11x + 3)
The exponents move out front
-7x ln (4) = (x + 3) ln (11)
the ln(4) and ln(11) are irrational numbers, we will leave them in this format until we simplify at the end. Use the distributive property on the right side
-7x ln (4) = x ln (11) + 3 ln (11)
Move all values with x to the left side
-7x ln (4) – x ln (11) = 3 ln (11)
Factor out a x on the left side
x (-7 ln (4) – ln (11)) = 3 ln (11)
Divide both sides by (7 ln (4) + ln (11))
x = [latex]\frac{3ln(11)}{-7ln(4)-ln(11)}[/latex]
x = -0.59