Properties of Logarithms MATH Precalculus S. Rook

Overview Section 3.3 in the textbook: – Properties of logarithms – Change-of-base formula – Logarithmic scales 2

Properties of Logarithms

4 Logarithms can be manipulated using a set of very important properties: – Product: log a (uv) = log a u + log a v NOTE: – Quotient: log a ( u ⁄ v ) = log a u – log a v NOTE: – Power: log a (u n ) = n ∙ log a u Applicable to logarithms with ANY valid base including common and natural logarithms The bases of the logarithms MUST be the same Used to write equivalent logarithmic expressions

5 Expanding & Compressing Logarithms Tips when expanding one logarithm into multiple logarithms with the SAME base as the original: – Work from outer to inner Tips when compressing several logarithms of the SAME base into one logarithm of that SAME base: – Apply the power property if necessary Removes coefficients from in front of logarithms Logarithms must NOT have a coefficient in front when combining – Work from inner to outer – Apply the product and quotient properties of logarithms to combine

Expanding Logarithms (Example) Ex 1: Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms: a)b) c) 6

Compressing Logarithms (Example) Ex 2: Condense the expression to the logarithm of a single quantity: a)b) c) 7

Change-of-Base Formula

9 Recall last lesson when we discussed that the calculator can only evaluate in base 10 (log) or base e (ln) – Also mentioned that we could “trick” the calculator into evaluating in other bases Change-of-Base Formula: – Note that the base in the ratios can be any value – just as long as it is the SAME base e.g.

Change-of-Base Formula (Example) Ex 3: Approximate the logarithm to three decimal places using the change-of-base formula with a) log b) ln: a) b) 10

Logarithmic Scales

12 Logarithmic Scales Used to scale very large or very small numbers to a more easily understood interval We will see this applied with the Richter Scale

13 Richter Scale Magnitude The Richter scale is used to convert earthquake intensities to a 0 to 10 scale – A logarithmic scale is required because the intensities can grow extremely large Because intensities are scaled down so compactly, the difference in intensities between any two numbers on the 0 to 10 scale is significant Richter Scale Magnitude:

14 Richter Scale Magnitude (Example) Ex 4: Compare the intensity of an earthquake that measured 4.5 on the Richter Scale with an earthquake that measured 5.5 on the Richter Scale

Summary After studying these slides, you should be able to: – Use the properties of logarithms to condense and expand logarithmic expressions – Apply the change-of-base formula for bases other than e or 10 – Solve application problems involving logarithmic scales Additional Practice – See the list of suggested problems for 3.3 Next lesson – Exponential & Logarithmic Equations (Section 3.4) 15