Section 6.6 - Exponential and Logarithmic Equations

Section Objectives

1. Solve exponential equations.
2. Solve logarithmic equations.
3. Solve exponential and logarithmic equations in applications.

Solving Exponential and Logarithmic Equations

In general, finding exact solutions of exponential or logarithmic equations can be very difficult (if not impossible). There are several common strategies we can try.

1. Some kinds of logarithmic and exponential equations can be solved by appealing to the one-to-one properties of logarithms and exponentials:

   $$a^x = a^y \quad \mbox{if and only if} \quad x=y,\\[0.1 in] \log_a x = \log_a y \quad \mbox{if and only if} \quad x = y.$$

2. Use logarithms and their properties to reduce an exponential equation to a simpler equation.

3. Use exponentials and their properties to reduce a logarithmic equation to a simpler equation.

4. Use algebraic techniques such a factoring, common denominators, etc.

If an exact solution can be found, some combination of the above strategies will probably be helpful.

Examples

• Solve for $x$: $\quad 2^{3x+1}=512$

• Solve for $x$: $\quad 5-e^x=0$

• Solve for $x$: $\quad e^{x^2}=e^{3x+4}$

• Solve for $x$: $\quad e^x+5=60$

• Solve for $x$: $\quad e^{2x}-7e^x+12=0$

• Solve for $x$: $\quad \log_6(3x+14)-\log_6 5 = \log_6 2x$

• Solve for $x$: $\quad 2 \ln 3x = 4$

• Solve for $x$: $\quad \log 5x + \log(x+1) = 2$

• An object at a temperature of $80^{\circ}$C is placed into a room at $20^{\circ}$C. The object cools so that its temperature after $t$ minutes is given by $T=20+60e^{-0.06t}$. How long does it take the object to reach $70^{\circ}$C?