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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

 

 

 

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