Describe the strategy you would use to solve \log_{6}x-\log_{6}4+\log_{6}8

coexpennan 2021-09-27 Answered
Describe the strategy you would use to solve \(\displaystyle{{\log}_{{{6}}}{x}}-{{\log}_{{{6}}}{4}}+{{\log}_{{{6}}}{8}}\).
a. Use the product rule to turn the right side of the equation into a single logarithm. Recognize that the resulting value is equal to x.
b. Express the equation in exponential form, set the exponents equal to each other and solve.
c. Use the fact that the logs have the same base to add the expressions on the right side of the equation together. Express the results in exponential form, set the exponents equal to each other and solve.
d. Use the fact that since both sides of the equations have logarithms with the same base to set the expressions equal to each other and solve.

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Expert Answer

izboknil3
Answered 2021-09-28 Author has 6736 answers
\(\displaystyle{{\log}_{{{6}}}{x}}={{\log}_{{{6}}}{4}}+{{\log}_{{{6}}}{8}}\)
\(\displaystyle{{\log}_{{{6}}}{x}}={{\log}_{{{6}}}{\left({32}\right)}}{\left\lbrace{{\log}_{{{c}}}{a}}+{{\log}_{{{c}}}{b}}={{\log}_{{{c}}}{\left({a}{b}\right)}}\right\rbrace}\) Product rule.
\(\displaystyle{x}={32}\)
Option A is correct.
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Relevant Questions

asked 2021-10-08

To solve:
\(\displaystyle{\left|{x}+{6}\right|}{>}{0}\)
General strategy to solve the inequalities that involve absolute value:
Absolute value inequalities deal with the inequalities \(\displaystyle{\left({<},{>},\leq,{\quad\text{and}\quad}\geq\right)}\) on the expressions with absolute sign.
General properties for solving inequalities that involve absolute value are
1) \(\displaystyle{\left|{x}\right|}{>}{k}\) is equivalent to \(\displaystyle{x}{<}-{k}\ {\quad\text{or}\quad}\ {x}{>}{k}\), where k is positive number.
2) \(\displaystyle{\left|{x}\right|}{<}{k}\) is equivalent to \(\displaystyle{x}\succ{k}\ {\quad\text{and}\quad}\ {x}{<}{k}\), where k is a positive number and we can write a conjunction such as \(\displaystyle{x}\succ{k}\ {\quad\text{and}\quad}\ {x}{<}{k}\) in the compact form \(\displaystyle-{k}{<}{x}{<}{k}\).
If k is a non positive number, we can determine the solution sets by inspection.

asked 2021-10-17
Solve the following logarithmic equations and inequalities:
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asked 2021-10-11

To solve:
\(\displaystyle{\left|{x}-{2}\right|}{>}{6}\)
General strategy to solve the inequalities that involve absolute value:
Absolute value inequalities deal with the inequalities \(\displaystyle{\left({<},{>},\leq,{\quad\text{and}\quad}\geq\right)}\) on the expressions with absolute sign.
We can use the property \(\displaystyle{\left|{x}\right|}{>}{k}\) is equivalent to \(\displaystyle{x}{<}-{k}\ {\quad\text{or}\quad}\ {x}{>}{k}\), where k is a positive number
When solving an absolute value inequality it's necessary to first isolate the absolure value expression on one side of the inequality before solving the inequality.
\(\displaystyle{\left|{a}{x}+{b}\right|}{<}{c}\), where \(\displaystyle{c}{>}{0}\)
\(\displaystyle=-{c}{<}{a}{x}+{b}{<}{c}\)
\(\displaystyle{\left|{a}{x}+{b}\right|}{>}{c}\), where \(\displaystyle{c}{>}{0}\)
\(\displaystyle={a}{x}+{b}{<}-{c}\ {\quad\text{or}\quad}\ {a}{x}+{b}{>}{c}\)
We can replace > above with \(\displaystyle\geq\ {\quad\text{and}\quad}{<}{w}{i}{t}{h}\leq\).

asked 2021-10-14

To solve:
\(\displaystyle{\left|{3}{x}+{1}\right|}\leq{13}\).
General strategy to solve the inequalities that involve absolute value:
Absolute value inequalities deal with the inequalities \(\displaystyle{\left({<},{>},\leq,{\quad\text{and}\quad}\ \geq\right)}\) on the expressions with absolute sign.
We can use the property \(\displaystyle{\left|{x}\right|}{<}{k}\) is equivalent to \(\displaystyle{x}\succ{k}\ {\quad\text{and}\quad}\ {x}{<}{k}\), where k is a positive number and we can write a conjuction such as \(\displaystyle{x}\succ{k}\ {\quad\text{and}\quad}\ {x}{<}{k}\) in the compact form.
\(\displaystyle-{k}{<}{x}{<}{k}\).
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\(\displaystyle{\left|{x}\right|}{<}{2}\), represents the distance between x and 0 that is less than 2.
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We can write an absolute value inequality as a compound inequality \(\displaystyle{\left({i}.{e}.\right)}-{2}{<}{x}{<}{2}\).
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\(\displaystyle{\left|{a}{x}+{b}\right|}{<}{c}\), where \(\displaystyle{c}{>}{0}\)
\(\displaystyle=-{c}{<}{a}{x}+{b}{<}{c}\)
\(\displaystyle{\left|{a}{x}+{b}\right|}{>}{c}\), where \(\displaystyle{c}{>}{0}\)
\(\displaystyle={a}{x}+{b}{<}-{c}\ {\quad\text{or}\quad}\ {a}{x}+{b}{>}{c}\)
We can replace > above with \(\displaystyle\geq\ {\quad\text{and}\quad}\ {<}\ {w}{i}{t}{h}\ \leq\).

asked 2021-10-08

To solve:
\(\displaystyle{\left|{3}{x}-{1}\right|}\succ{2}\)
General strategy to solve the inequalities that involve absolute value:
Absolute value inequalities deal with the inequalities \(\displaystyle{\left({<},{>},\leq,{\quad\text{and}\quad}\geq\right)}\) on the expressions with absolute sign.
General properties for solving inequalities that involve absolute value are
1) \(\displaystyle{\left|{x}\right|}{>}{k}\) is equivalent to \(\displaystyle{x}{<}-{k}\ {\quad\text{or}\quad}\ {x}{>}{k}\), where k is positive number.
2) \(\displaystyle{\left|{x}\right|}{<}{k}\) is equivalent to \(\displaystyle{x}\succ{k}\ {\quad\text{and}\quad}\ {x}{<}{k}\), where k is a positive number and we can write a conjunction such as \(\displaystyle{x}\succ{k}\ {\quad\text{and}\quad}\ {x}{<}{k}\) in the compact form \(\displaystyle-{k}{<}{x}{<}{k}\).
If k is a non positive number, we can determine the solution sets by inspection.

asked 2021-10-17

To find:
The given statement is true or false.
General strategy to solve the inequalities that involve absolute value:
Absolute value inequalities deal with the inequalities \(\displaystyle{\left({<},{>},\leq,{\quad\text{and}\quad}\geq\right)}\) on the expressions with absolute sign.
General properties for solving inequalities that involve absolute value are
1) \(\displaystyle{\left|{x}\right|}{>}{k}\) is equivalent to \(\displaystyle{x}{<}-{k}\ {\quad\text{or}\quad}\ {x}{>}{k}\), where k is positive number.
2) \(\displaystyle{\left|{x}\right|}{<}{k}\) is equivalent to \(\displaystyle{x}\succ{k}\ {\quad\text{and}\quad}\ {x}{<}{k}\), where k is a positive number and we can write a conjunction such as \(\displaystyle{x}\succ{k}\ {\quad\text{and}\quad}\ {x}{<}{k}\) in the compact form \(\displaystyle-{k}{<}{x}{<}{k}\).
If k is a non positive number, we can determine the solution sets by inspection.

asked 2021-09-29
Solve the below equations and inequalities for the given variable.
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