8.6 Solving Exponential and Logarithmic Equations

Solving by Equating Exponents One method of solving equations is to use the property that if two powers with the same base are equal, then their exponents must be equal. For b>0 and b≠1, if bx = by, then x = y

Examples Solve. 24x =32x-1 92x = 813x-1

Taking A logarithm of Each Side When it is not convenient to write each side of an exponential equation using the same base, you can solve the equation by taking a logarithm of each side.

Examples: Solve. 4x = 15 5x = 18 5x+2 + 3=25 8+105x+4 = 35

Example: Newton’s law of cooling states that the temperature T of a cooling substance at time t(in minutes) can be modeled by the equation T= (T0 – TR)e-rt + TR , where T0 is the initial temperature of the substance, TR is the room temperature, and r is a constant that represents the cooling rate of the substance.

Example (Continue) How long will it take a liquid to cool to a temperature of 90oF, if room temperature isb70oF and the cooling rate is r=0.046.

Solving a Logarithmic Equations To solve a logarithmic equation, use the property for logarithms with the same base: For positive numbers b, x, and y where b≠1, logbx = logby if and only if x = y

Example Solve log4 (x+3) = log4(8x+17) log2 (2x-1) = log2 (x+5)

Exponentiate each side When it is not convenient to write both sides of an equation as logarithmic expression with the same base, you can exponentiate each side of the equation. For b>0 and b1, if x = y, the bx = by

Example Solve log2 (2x-1) = log2 (x+5) log5 (x-4) = 2

Extraneous Solutions Because the domain of a logarithmic function does not include all real numbers, you should be sure to check for extraneous(not valid) solutions of the logarithmic equations.

Examples: Solve. log2 x + log2 (x-7) = 3 log6 (x+5) + log6x = 2

Example: The moment magnitude M of an earthquake that releases energy E(in ergs) can be modeled by the equation M = 0.291 ln E + 1.17. If the earthquake in Prince William Sound in 1964 had a moment magnitude of 8.6, how much energy did it release?

Example: Solve 14 = 8e0.02t