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Answer :
Sure! Let's go through the process of finding [tex]\(\log_{18} 124\)[/tex].
To find the logarithm of a number with a specific base, you can use the change of base formula. The change of base formula states:
[tex]\[
\log_b a = \frac{\log_c a}{\log_c b}
\][/tex]
In this formula:
- [tex]\(b\)[/tex] is the base of the logarithm you are interested in (18 in this case).
- [tex]\(a\)[/tex] is the number you want to find the logarithm of (124 in this case).
- [tex]\(c\)[/tex] is any positive number you choose as the new base for your calculations. A common choice is base 10 (common logarithms) or base [tex]\(e\)[/tex] (natural logarithms), but any positive base can be used.
We'll use the natural logarithm (base [tex]\(e\)[/tex]), which is typically denoted as [tex]\(\ln\)[/tex].
Applying the change of base formula to our problem:
[tex]\[
\log_{18} 124 = \frac{\ln 124}{\ln 18}
\][/tex]
By calculating [tex]\(\ln 124\)[/tex] and [tex]\(\ln 18\)[/tex] separately using a calculator, and then dividing the two results, you get:
1. Calculate [tex]\(\ln 124\)[/tex].
2. Calculate [tex]\(\ln 18\)[/tex].
3. Divide the value from step 1 by the value from step 2:
[tex]\[
\frac{\ln 124}{\ln 18} \approx 1.6677029701928758
\][/tex]
Thus, [tex]\(\log_{18} 124 \approx 1.6677\)[/tex].
This means that 18 raised to the power of approximately 1.6677 equals 124.
To find the logarithm of a number with a specific base, you can use the change of base formula. The change of base formula states:
[tex]\[
\log_b a = \frac{\log_c a}{\log_c b}
\][/tex]
In this formula:
- [tex]\(b\)[/tex] is the base of the logarithm you are interested in (18 in this case).
- [tex]\(a\)[/tex] is the number you want to find the logarithm of (124 in this case).
- [tex]\(c\)[/tex] is any positive number you choose as the new base for your calculations. A common choice is base 10 (common logarithms) or base [tex]\(e\)[/tex] (natural logarithms), but any positive base can be used.
We'll use the natural logarithm (base [tex]\(e\)[/tex]), which is typically denoted as [tex]\(\ln\)[/tex].
Applying the change of base formula to our problem:
[tex]\[
\log_{18} 124 = \frac{\ln 124}{\ln 18}
\][/tex]
By calculating [tex]\(\ln 124\)[/tex] and [tex]\(\ln 18\)[/tex] separately using a calculator, and then dividing the two results, you get:
1. Calculate [tex]\(\ln 124\)[/tex].
2. Calculate [tex]\(\ln 18\)[/tex].
3. Divide the value from step 1 by the value from step 2:
[tex]\[
\frac{\ln 124}{\ln 18} \approx 1.6677029701928758
\][/tex]
Thus, [tex]\(\log_{18} 124 \approx 1.6677\)[/tex].
This means that 18 raised to the power of approximately 1.6677 equals 124.
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