Express the following logarithms in terms of ln5 and ln7 a. ln (1 / 125) b. ln 9.8

Express the following logarithms in terms of ln5 and ln7 ...

Express the following logarithms in terms of $\ln 5$ and $\ln 7$
$$
\begin{array}{ll}{\text { a. } \ln (1 / 125)} & {\text { b. } \ln 9.8} \\ {\text { d. } \ln 1225} & {\text { e. } \ln 0.056} \\ {\text { f. }(\ln 35+\ln (1 / 7)) /(\ln 25)}\end{array}
$$

Key Concepts

Properties of Logarithms

Logarithms have several fundamental properties that allow for the manipulation of expressions. The product rule states that the logarithm of a product is the sum of the logarithms; the quotient rule states that the logarithm of a quotient is the difference of the logarithms; and the power rule allows exponents to be brought out in front of the log as a multiplicative factor. These properties are essential in rewriting expressions into forms involving specific logarithms.

Power Rule for Logarithms

The power rule for logarithms, expressed as ln(a^b) = b * ln(a), is instrumental when numbers are represented as exponents. By recognizing and applying this rule, one can pull the exponent out in front of the logarithm, which simplifies the expression into terms that can be directly related to given logarithmic values.

Prime Factorization

Expressing numbers in terms of their prime factors is a key strategy when converting logarithmic expressions to a desired form. By decomposing a composite number into primes (for example, rewriting a number as a power of 5 or 7), one can apply logarithmic properties to express the result solely in terms of the logarithms of those prime numbers.

Expressing Logarithms in Terms of Known Values

Sometimes the task is to express a logarithmic expression using a restricted set of known logarithmic values. This concept involves strategically rewriting and decomposing numbers so that the final expression is composed only of the logarithms of the given numbers. This technique typically relies on the aforementioned properties and factorization methods.

Transcript

00:01 This problem, we're asked to write these various expressions that are given in terms of the natural log of 5 and the natural log of 7.

00:11 So in this problem, it's really more a matter of figuring out how to write what's inside of this, the natural log here, as these as values in terms of 5 and 7.

00:24 So this one is actually, you know, we can write it as the natural log of 1 minus the natural log of 125.

00:32 Natural log of 1 is 0.

00:36 It is 5 cubed.

00:38 And so using the formula for, with natural logs, this winds up becoming minus 3 times the natural log of 5.

00:47 So this is minus 3 times the natural log of 5.

00:50 Now the natural log of 8 or 9 .8, well, 9 .8 is 49 over 5, which we can write as the natural log of that, we can write to the natural log of 49 minus the natural log of 5.

01:04 Is obviously 7 squared, so that becomes a natural log of 7 minus the natural log of 5.

01:12 Now, natural log of 7 times the square root of 7, we can write as the natural log of 7 plus the natural log of square root is 7.

01:21 The natural log of square root of 7 is the natural log of 7 to the 1 1⁄2, which is the natural log of 1⁄2 times the natural log of 7, which is the natural 0⁄2.

01:31 Which is the natural log of 7.

01:33 We could have done that immediately because 7 squared 2.

01:36 Over to seven is seven to three halves.

01:39 And then we could have just jumped pretty much directly to there.

01:46 1 ,225 is 49 times 25.

01:52 So the natural log of that is the natural log of 49 plus the natural log of 25...

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