Using Exponentials and Logs

Using Exponentials and Logs

Exponential Functions

Understanding Exponential Functions

  • An exponential function has a variable in the exponent and can be written in the form f(x) = a^x, where a is a positive constant and x is a variable.

  • Exponential functions grow by equal factors over equal intervals. This is also known as exponential growth.

  • The base of the exponential function is the constant a. If a > 1, it shows exponential growth and if 0 < a < 1, it shows exponential decay.

Graphing Exponential Functions

  • The graph of an exponential function f(x) = a^x always lies above the x-axis but can get arbitrarily close to it. It is asymptotic to the x-axis.

  • For a > 1, the graph of an exponential function rises to the right and falls to the left. For 0 < a < 1, the function rises to the left and falls to the right.

Properties of Exponential Functions

  • a^0 = 1 and a^n / a^m = a^{n-m}

  • (a^b)^c = a^{bc}: Exponents multiply when they are within an exponent.

  • a^b * a^c = a^{b+c}: Exponents add when multiplying terms with the same base.

Logarithmic Functions

Understanding Logarithmic Functions

  • A logarithm is a way to find the power needed to raise a certain base number to get a certain result.

  • In other words, if a^x = b then log_a b = x.

  • The base of a logarithm is always a positive number, not equal to 1.

Properties of Logarithms

  • log_b(mn) = log_b(m) + log_b(n): Logarithms of products are the sum of the logarithms.

  • log_b(m/n) = log_b(m) - log_b(n): Logarithms of quotients are the difference of the logarithms.

  • log_b(m)^n = n log_b(m): The logarithm of a power of a number is the product of the logarithm of the number and the power.

Solving Logarithmic Equations

  • A logarithmic equation is an equation that involves the logarithm of an expression containing a variable.

  • To solve logarithmic equation, use the principles of logarithms to simplify the equation and then solve for the unknown.

Converting Between Logarithms and Exponentials

  • Logarithmic and exponential expressions are able to be converted between each other using their definitions.

  • To convert an exponential expression to logarithmic form or vice versa, identify the base, exponent and result in the exponential form and rearrange accordingly.

Remember to use logarithmic tables or scientific calculators when necessary to compute values involving logarithms or exponential functions.