Question

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

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

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,
$$\displaystyle{8}{x}-{16}-{x}={x}+{5}-{x}$$
$$\displaystyle{7}{x}-{16}={5}$$
$$\displaystyle{7}{x}-{16}+{16}={5}+{16}$$
$$\displaystyle{7}{x}={21}$$
$$\displaystyle{\frac{{{7}{x}}}{{{7}}}}={\frac{{{21}}}{{{7}}}}$$
$$\displaystyle{x}={3}$$ Thus, the exact solution is $$\displaystyle{x}={3}.$$