Logarithms

*General equation for a transformed log function

y=a log [k(x-d)]+c

*Things to remember when solving logarithmic equations

1. Comparing arguments (see photo) 2. Always watch for inadmissible solutions when you are working with logarithms 3. To rewrite a logarithm in exponential form, remember: (see photo)

Laws of logarithms

1. Product law 2. Quotient law 3. Power law

Steps to solving logarithmic equations

1. Simplify as much as you can using basic algebra 2. Use log laws when you need to so that you end up with a single logarithm 3. Write the single logarithm in its exponential form 4. Solve the exponential equation

Methods for solving exponential equations

1. Write both sides with the same base and compare exponents 2. Take the log of both sides and apply rules of logs to solve

What does "k" represent

horizontal stretch/compression by a factor of "1/k"

What is the parent function of the log function

logx (base 10 is assumed→log₁₀x)

Does log -x exist? Why?

no because there is a vertical asymptote at x=0, therefore the domain is {x∈R|x>0}

What does "c" represent

vertical shift up (+c) or down (-c)

What does "a" represent

vertical stretch/compression by a factor of "a"

Helpful hints when evaluating logarithms

-logarithms of negative numbers DON'T exist (we can't write a negative number as a power of a positive base) -we can estimate values of logarithms by guessing and checking with our calculator -most often, rewriting a logarithmic equation in exponential form will allow us to solve for our variable

Characteristics of logarithmic functions

-mirror image of the exponential function in the line y=x -passes through (1,0) -switch x and y coordinates of exponential functions -V.A. at x=0

*Inadmissible solutions when solving logarithms

-the base must always be a positive number -the argument, or the expression you are taking of the logarithm, must always be positive see photo for example

Annually vs. semi-annually vs. quarterly

Determines what you divide the interest rate by i (as a decimal)/number of times compounded in a year annually: 1 semi-annually: 2 quarterly: 4 weekly: 52 monthly: 12`

Formula involving decibels: L=10 log I/Io What does each letter represent?

L=loudness of a sound in decibels (dB) I=intensity of the sound in watts per square meter (****W/m²)****) Io=10⁻¹² W/m²

What does a, x and y represent in the general equations of the last card?

a=base x=argument (input) y=output

What is true for both exponential and logarithmic functions

a>0 and a≠1 -if a was equal to one the function would be a straight line

What is the purpose of the log function

acts as the inverse of the exponential function and undoes the exponential operation

common logarithm

an expression in the form log x; this is assumed to have a base of 10 and can be evaluated using your calculator button

What does "d" represent

horizontal shift -(x-d)=+d→shift right -(x+d)=-d→shift left