Exponential and Logarithmic Equations

Exponential Equations

• Solve using the one-to-one property: If $b^s = b^t$, then $s = t$ (where $b > 0$ and $b \neq 1$).

  • Example: $4^{-3v-2} = 4^{-v} \implies -3v - 2 = -v$

• If bases are not the same, rewrite them to have the same base.

  • Example: $81^x = 3^{2x} \implies (3^4)^x = 3^{2x} \implies 3^{4x} = 3^{2x} \implies 4x = 2x$

• When multiplying exponential expressions with the same base, add the exponents:

  • Example: $3^{2x+1} \cdot 3^x = 243 \implies 3^{3x+1} = 3^5$

Solving Exponential Equations Using Logarithms

• Isolate the exponential expression.

• Take the logarithm (log or ln) of both sides.

• Use the power property of logarithms to bring down exponents.

• Solve for the variable.

• Example: $4^{3-2x} = 7 \implies (3-2x)\log(4) = \log(7)$

Solving Exponential Equations with Different Bases Using Logarithms

• Isolate the exponential expression on one side.

• Take the logarithm of both sides.

• Use the power property of logarithms.

• Solve for the variable.

Solving Logarithmic Equations

• Isolate logarithmic terms.

• Condense into one logarithm.

• Rewrite in exponential form.

• Solve for the variable.

• Remember to check for extraneous solutions.

Extraneous Solutions

• Solutions that satisfy the algebraic steps but not the original equation.

• Logarithms of negative numbers are undefined, which can lead to extraneous solutions.

  • Example: $\ln(x) = -1$ has no solution because $x$ would be negative.

l# Continuous Compound Interest

• Formula: $A = Pe^{rt}$, where:

  • $A$ is the final amount

  • $P$ is the principal amount

  • $r$ is the interest rate

  • $t$ is the time in years

• Example to find doubling time:

  • $5000 = 2500e^{0.045t} \implies 2 = e^{0.045t} \implies t = \frac{\ln(2)}{0.045}$