Logarithmic Expressions and Equations

Logarithmic expressions are a way of expressing the power to which a base number must be raised to achieve a certain result. Logarithmic equations, on the other hand, require finding the value of an exponent-based variable. These concepts are not only crucial in areas such as physics, engineering, and computer science, but they are also solved by employing logarithmic properties or converting them into exponential equations.

Simplifying Logarithmic Expressions

te the expression as a single logarithm. \[ \frac{1}{3} \ln (x+2)^{3}+\frac{1}{2}\left[\ln (x)-\ln \left(x^{2}+3 x+2\right)^{2}\right] \]
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Expanding Logarithmic Expressions

Solve the logarithmic equation. Be sure to reject any value of $x$ that is not in the domain of the original logarithmic expressions. Give the exact answer. \[ \log _{5}(x+7)+\log _{5}(x+3)=1 \] Rewrite the given equation without logarithms. Do not solve for $x$.
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Evaluating Logarithms

If $\log (x)=-0.123$, what does $x$ equal? Express your answer numerically using three significant figures. View Available Hint(s)
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Rewriting in Exponential Form

Solve the logarithmic equation \(\log_2(x) = 5\).
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Converting to Logarithmic Form

Express the equation in logarithmic form: (a) $4^{3}=64$ is equivalent to $\log A=B$. \[ A= \] and \[ B= \] (b) $10^{-4}=0.0001$ is equivalent to $\log _{10} C=D$. \[ C= \] and \[ D= \]
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