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Express each logarithm in terms of [tex]\ln 3[/tex] and [tex]\ln 5[/tex].

A. [tex]\ln \frac{81}{125}[/tex]

B. [tex]4 \ln 5 - 3 \ln 3[/tex]

C. [tex]5 \ln 3 - 3 \ln 4[/tex]

D. [tex]4 \ln 3 - 3 \ln 5[/tex]

E. [tex]3 \ln 4 - 5 \ln 3[/tex]


Sagot :

Sure! Let's take the expression [tex]\(\ln \frac{81}{125}\)[/tex] and express it in terms of [tex]\(\ln 3\)[/tex] and [tex]\(\ln 5\)[/tex]:

1. Rewrite as a difference of logarithms:

[tex]\[ \ln \frac{81}{125} = \ln 81 - \ln 125 \][/tex]

2. Express the arguments as powers:

Notice that [tex]\(81 = 3^4\)[/tex] and [tex]\(125 = 5^3\)[/tex]:
[tex]\[ \ln 81 = \ln (3^4) \quad \text{and} \quad \ln 125 = \ln (5^3) \][/tex]

3. Apply the power rule of logarithms:

[tex]\(\ln (a^b) = b \ln a\)[/tex], so:
[tex]\[ \ln (3^4) = 4 \ln 3 \quad \text{and} \quad \ln (5^3) = 3 \ln 5 \][/tex]

4. Substitute these back into the original expression:

[tex]\[ \ln \frac{81}{125} = 4 \ln 3 - 3 \ln 5 \][/tex]

Thus, [tex]\(\ln \frac{81}{125}\)[/tex] can be expressed in terms of [tex]\(\ln 3\)[/tex] and [tex]\(\ln 5\)[/tex] as:

[tex]\[ \ln \frac{81}{125} = 4 \ln 3 - 3 \ln 5 \][/tex]

Given the expressions provided, the correct one is:
[tex]\[ 4 \ln 3 - 3 \ln 5 \][/tex]

So, the correct expression in terms of [tex]\(\ln 3\)[/tex] and [tex]\(\ln 5\)[/tex] is:
[tex]\[ 4 \ln 3 - 3 \ln 5 \][/tex]