6.6 Exponential and logarithmic equations (Page 2/8)

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Rewriting equations so all powers have the same base

Sometimes the common base for an exponential equation is not explicitly shown. In these cases, we simply rewrite the terms in the equation as powers with a common base, and solve using the one-to-one property.

For example, consider the equation $256 = 4^{x - 5} .$ We can rewrite both sides of this equation as a power of $2.$ Then we apply the rules of exponents, along with the one-to-one property, to solve for $x :$

\begin{array}{l} 256 = 4^{x - 5} \\ 2^{8} = {(2^{2})}^{x - 5} & Rewrite each side as a power with base 2 . \\ 2^{8} = 2^{2 x - 10} & Use the one-to-one property of exponents . \\ 8 = 2 x - 10 \begin{matrix}  \end{matrix} & Apply the one-to-one property of exponents . \\ 18 = 2 x & Add 10 to both sides . \\ x = 9 & Divide by 2 . \end{array}

How To

Given an exponential equation with unlike bases, use the one-to-one property to solve it.

Rewrite each side in the equation as a power with a common base.
Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form $b^{S} = b^{T} .$
Use the one-to-one property to set the exponents equal.
Solve the resulting equation, $S = T,$ for the unknown.

Solving equations by rewriting them to have a common base

Solve $8^{x + 2} = 16^{x + 1} .$

\begin{array}{l} 8^{x + 2} = 16^{x + 1} \\ {(2^{3})}^{x + 2} = {(2^{4})}^{x + 1} & Write 8 and 16 as powers of 2. \\ 2^{3 x + 6} = 2^{4 x + 4} & To take a power of a power, multiply exponents . \\ 3 x + 6 = 4 x + 4 & Use the one-to-one property to set the exponents equal . \\ x = 2 & Solve for x . \end{array}

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Solve $5^{2 x} = 25^{3 x + 2} .$

$x = - 1$

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Solving equations by rewriting roots with fractional exponents to have a common base

Solve $2^{5 x} = \sqrt{2} .$

\begin{array}{l} 2^{5 x} = 2^{\frac{1}{2}} & Write the square root of  2 as a power of 2. \\ 5 x = \frac{1}{2} & Use the one-to-one property . \\ x = \frac{1}{10} & Solve for x . \end{array}

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Solve $5^{x} = \sqrt{5} .$

$x = \frac{1}{2}$

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Q&A

Do all exponential equations have a solution? If not, how can we tell if there is a solution during the problem-solving process?

No. Recall that the range of an exponential function is always positive. While solving the equation, we may obtain an expression that is undefined.

Solving an equation with positive and negative powers

Solve $3^{x + 1} = −2.$

This equation has no solution. There is no real value of $x$ that will make the equation a true statement because any power of a positive number is positive.

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Solve $2^{x} = −100.$

The equation has no solution.

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Solving exponential equations using logarithms

Sometimes the terms of an exponential equation cannot be rewritten with a common base. In these cases, we solve by taking the logarithm of each side. Recall, since $\log (a) = \log (b)$ is equivalent to $a = b,$ we may apply logarithms with the same base on both sides of an exponential equation.

How To

Given an exponential equation in which a common base cannot be found, solve for the unknown.

Apply the logarithm of both sides of the equation.
- If one of the terms in the equation has base 10, use the common logarithm.
- If none of the terms in the equation has base 10, use the natural logarithm.
Use the rules of logarithms to solve for the unknown.

Solving an equation containing powers of different bases

Solve $5^{x + 2} = 4^{x} .$

\begin{array}{l} 5^{x + 2} = 4^{x} & There is no easy way to get the powers to have the same base . \\ \ln 5^{x + 2} = \ln 4^{x} & Take ln of both sides . \\ (x + 2) \ln 5 = x \ln 4 & Use laws of logs . \\ x \ln 5 + 2 \ln 5 = x \ln 4 & Use the distributive law . \\ x \ln 5 - x \ln 4 = - 2 \ln 5 & Get terms containing x on one side, terms without x on the other . \\ x (\ln 5 - \ln 4) = - 2 \ln 5 & On the left hand side, factor out an x . \\ x \ln (\frac{5}{4}) = \ln (\frac{1}{25}) \begin{matrix}  \end{matrix} & Use the laws of logs . \\ x = \frac{\ln (\frac{1}{25})}{\ln (\frac{5}{4})} & Divide by the coefficient of x . \end{array}

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Questions & Answers

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advantages of electrons in a circuit

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Source: OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6

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