Question

Solve exponential and logarithmic equations \log_{8}(x+5)-\log_{8}(x-2)=1

Decimals
ANSWERED
asked 2021-08-06
Exponential and Logarithmic Equations Solve the equation. Find the exact solution if possible. otherwise, use a calculator to approximate to two decimals.
\(\displaystyle{{\log}_{{{8}}}{\left({x}+{5}\right)}}-{{\log}_{{{8}}}{\left({x}-{2}\right)}}={1}\)

Answers (1)

2021-08-07
Step 1
Approach:
For solving Exponential Equations:
1: Isolate the exponential expression on one side of the equation.
2: Take the logarithm of each side, then use the Laws of Logarithms to "bring down exponent."
3: Solve for the variable.
Step 2
Given,
\(\displaystyle{{\log}_{{{8}}}{\left({x}+{5}\right)}}-{{\log}_{{{8}}}{\left({x}-{2}\right)}}={1}\)
By the second law of logarithms,
\(\displaystyle{{\log}_{{{a}}}{\left({\frac{{{A}}}{{{B}}}}\right)}}={{\log}_{{{a}}}{A}}-{{\log}_{{{a}}}{B}}\)
\(\displaystyle{{\log}_{{{8}}}{\frac{{{\left({x}+{5}\right)}}}{{{\left({x}-{2}\right)}}}}}={1}\)
By the definition of logarithm function,
\(\displaystyle{{\log}_{{{a}}}{x}}={y}\Leftrightarrow\ {a}^{{{y}}}={x}\)
\(\displaystyle{8}^{{{1}}}={\frac{{{\left({x}+{5}\right)}}}{{{\left({x}-{2}\right)}}}}\)
\(\displaystyle{8}={\frac{{{\left({x}+{5}\right)}}}{{{\left({x}-{2}\right)}}}}\)
Multiply by \(\displaystyle{\left({x}-{2}\right)}\) on both sides,
\(\displaystyle{8}{\left({x}-{2}\right)}={\frac{{{\left({x}+{5}\right)}}}{{{\left({x}-{2}\right)}}}}{\left({x}-{2}\right)}\)
\(\displaystyle{8}{\left({x}-{2}\right)}={\left({x}+{5}\right)}\)
By the distributive property of multiplication,
\(\displaystyle{8}{\left({x}\right)}-{8}{\left({2}\right)}={\left({x}+{5}\right)}\)
\(\displaystyle{8}{x}-{16}={x}+{5}\)
Add \(\displaystyle{\left(-{x}\right)}\) on both sides,
\(\displaystyle{8}{x}-{16}-{x}={x}+{5}-{x}\)
\(\displaystyle{7}{x}-{16}={5}\)
Add 16 on both sides,
\(\displaystyle{7}{x}-{16}+{16}={5}+{16}\)
\(\displaystyle{7}{x}={21}\)
Divide by 7 on both sides,
\(\displaystyle{\frac{{{7}{x}}}{{{7}}}}={\frac{{{21}}}{{{7}}}}\)
\(\displaystyle{x}={3}\) Thus, the exact solution is \(\displaystyle{x}={3}.\)
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