Question

Find the value of:
(1) \[{{\log }_{_{\dfrac{1}{2}}}}8\]
(2) \[{{\log }_{5}}0.008\]
(3) \[{{\log }_{5}}3125\]
(4) \[{{\log }_{7}}\sqrt[3]{7}\]

Answer 1

Hint: If x and y are positive real numbers and b do not equal 1, then \[{{\log }_{b}}(x)=y\] is equivalent to \[{{b}^{y}}=x\]. Using this property we will solve this problem.
Complete step-by-step answer:
1) Rewrite as an equation.
\[{{\log }_{\dfrac{1}{2}}}(8)=x\]
Rewrite \[{{\log }_{\dfrac{1}{2}}}(8)=x\] in exponential form using the definition of a logarithm
\[{{\left( \dfrac{1}{2} \right)}^{x}}=8\]
Create expressions in the equation that all have equal bases.
\[{{\left( {{2}^{-1}} \right)}^{x}}={{2}^{3}}\]
Rewrite \[{{\left( {{2}^{-1}} \right)}^{x}}\]as \[\left( {{2}^{-x}} \right)\]
\[{{2}^{-x}}={{2}^{3}}\]
Since the bases are the same, then two expressions are only equal if the exponents are also equal.
\[-x=3\]
Solve for x.
\[x=-3\]
The variable x is equal to -3.
(2) Rewrite as an equation.
\[{{\log }_{5}}0.008\] = x
Rewrite \[{{\log }_{5}}0.008\] in exponential form using the definition of a logarithm
\[\left( {{5}^{x}} \right)=0.008\]
Create expressions in the equation that all have equal bases.
\[\left( {{5}^{x}} \right)=\dfrac{1}{125}\]
Rewrite \[\dfrac{1}{125}\]as \[\left( {{5}^{-3}} \right)\]
\[{{5}^{x}}={{5}^{-3}}\]
Since the bases are the same, then two expressions are only equal if the exponents are also equal.
\[x=-3\]
The variable x is equal to -3.
(3) Rewrite as an equation.
\[{{\log }_{5}}3125\] = x
Rewrite \[{{\log }_{5}}3125\] in exponential form using the definition of a logarithm
\[\left( {{5}^{x}} \right)=3125\]
Create expressions in the equation that all have equal bases.
\[{{5}^{x}}={{5}^{5}}\]
Since the bases are the same, then two expressions are only equal if the exponents are also equal.
\[x=5\]
The variable x is equal to 5.
(4) Rewrite as an equation.
\[\log _{7}{{{7}^{\dfrac{1}{3}}}}\] = x
Rewrite \[\log _{7}{{{7}^{\dfrac{1}{3}}}}\] in exponential form using the definition of a logarithm
Create expressions in the equation that all have equal bases.
\[{{7}^{x}}={{7}^{\dfrac{1}{3}}}\]
Since the bases are the same, then two expressions are only equal if the exponents are also equal.
\[x=\dfrac{1}{3}\]
The variable x is equal to \[\dfrac{1}{3}\].

Note: Log can take only positive values. To find the value of the above function, the function is converted to exponential form and further solved for the value of x. If the bases are the same, then two expressions are only equal if the exponents are also equal.