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Select all expressions that are equal to log₂ 8.
A. log₅ 20.
B. log₁₀ 125.
C. log₁₀ 100.
D. log₁₀ 1,000.
E. log₃ 27.
F. log₁₀ 0.001.

Sagot :

Answer:

D. log₁₀ 1,000.

E. log₃ 27

Step-by-step explanation:

Logarithms

Logarithms are the inverse equations of their exponential counterparts.

In an exponential equation, it finds the value of an exponent made a base and power.

In a logarithm equation, it finds the value of the power that produces an exponent with a certain base.

In other words,

                                                [tex]b^x=c[/tex]

                                           [tex]log_b(c)=x[/tex].

Knowing that logs are inverses of exponents can help evaluate them.

[tex]\hrulefill[/tex]

Solving the Problem

To determine the correct answers, we must find the value of [tex]log_2=8[/tex].

This logarithm means that when 2 is raised to some power x, it produces 8.

                                                  [tex]2^x=8[/tex]

Knowing 8 to be the cube value of 2,

                                         [tex]x=3 \rightarrow log_2(8)=3[/tex].

Now, we have to find answer choices where their value is also 3.

[tex]\dotfill[/tex]

Narrowing Down the Answer Choices

Using the facts mentioned in the top section, we can rewrite all of the logarithms as exponents.

[tex]A)\: 5^x=20\\\\B)\: 10^x=125\\\\C)\: 10^x=100\\\\[/tex]

[tex]D)\: 10^x=1000\\\\E)\: 3^x=27\\\\F)\: 10^x=0.001\\[/tex]

No power that raises 5 will get 20, so we can eliminate A.

The same can be said for B.

100 is the squared value of 10, meaning that x = 2. This isn't the same as the solution x = 3, so C is crossed off.

1000 is the cubed value of 10, meaning x = 3. So, D is a correct answer.

Same thing with E, 27 is the cubed value of 3, so E is also correct.

0.001 or [tex]\dfrac{1}{1000}[/tex] is the same as [tex]10^-^3[/tex], meaning x = -3. Not 3, so F is wrong.

Thus, D and E are our final answers.

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