Using the Compound Interest Formula and we’ll assume compound annually, we have:ttmt(1.05)3.051P3Pmr1PA

IssueThis equation would be easier to solve if the variable t was not in the exponent. In order to help us answer this question, we will need logarithms.

LogarithmFor a > 0, a ≠ 1, and x > 0,Ex.: The exponential form of 3² = 9 can be written in its logarithmic form of 2 = .xaxyyameanslog9log3

Logarithmic FunctionIf a > 0, a ≠ 1, then the logarithmic function of base a is defined byfor x > 0 log)(xxfa

One-to-one property for LogarithmIf a > 0, a ≠ 1, and , then x = y.ylogxlogaa

Inverse functionsLet’s look closely at the graphs of and f(2) = 9 where as g(9) = 2f(4) = 81 where as g(81) = 4f(0) = 1 where as g(1) = 0If a function has a point (x, y) and another function has a point (y, x), then these are considered as inverse functions. All inverse functions are reflected by the line x = y (the origin).xxf3)(xxg3log)(

Logarithmic Properties1.2.3.4.5.6. yxxyaaalogloglogyxyxaaalogloglogxrxaralog*log1logaa01logararalog

Common & Natural LogarithmsThe common logarithm is if we are taking the log to base 10 of x, this can be simplified as the following:Most applications of logarithms use the number e (2.7182818…). Logarithms to base e are called natural logarithms andxxloglog10xxelnlog

Computing other logarithms besides log x and ln x (Change of base)Any calculator can only compute the log to base 10 of x (log x) or the log to base e of x (ln x). But, it can’t compute something like . However, we have the following:Change of base Theorem for logarithmsIf x is any positive number and if a and b are positive real numbers, a ≠ 1, b ≠ 1, then (we will highly make b = 10 or e)44log3axxbbalogloglog

Another Change of BaseFor any positive numbers a and x, a ≠ 1, thenBack to example:axxalnlnlog445.33ln44ln44log445.33log44log3log44log44log310103

Logarithmic EquationsEquations involving logarithms are often solved by using the fact that exponential functions and logarithmic functions are inverses, so a logarithmic function can be rewritten as an exponential function. Other cases, the properties of logarithms may be useful in simplifying a logarithmic equation.

Example2,10)2)(1(02317)5(351731517log1)5(log)17(log222123323xxxxxxxxxxxxx

Note about Finding the solutions of a Logarithmic EquationWhen the original function is a logarithmic function, we should check the final answer(s) by plugging them into the original function. If we get the logarithm of a negative number or 0, then the solution is an extraneous solution.Example: Plugging -1 into each x (in the original question) is fine since both logarithms are positive. However, plugging in -6 to each into each x gives us a logarithm of a negative number which is not possible. Hence, the only acceptable answer is x = -11,60)1)(6(06710722)107(log2)2(log)5(log2222222xxxxxxxxxxxx

Inverse Properties for Exponential and Logarithmic functionsBefore seeing how we can solve exponential functions, it would be nice to see the following:g(x)allfor )()ln(exallfor )ln(e0g(x)for )(e0for x g(x)xln(g(x)))ln(xgxxgxex