Use the properties of logarithms to approximate the...

Use the properties of logarithms to approximate the indicated logarithms, given that $\ln 2 \approx 0.6931$ and $\ln 3 \approx 1.0986.$
(a) $\ln 0.25$
(b) $\ln 24$
(c) $\ln \sqrt[3]{12}$
(d) $\ln \frac{1}{72}$

Key Concepts

Natural Logarithm

The natural logarithm (ln) is the logarithm with base e, where e is an irrational mathematical constant approximately equal to 2.71828. It is a fundamental function in calculus and appears in various contexts, such as growth and decay processes, integration, and differential equations.

Properties of Logarithms

Logarithmic properties allow us to simplify complex logarithmic expressions into sums, differences, or multiples of simpler logarithms. Key properties include the product rule (ln(ab) = ln a + ln b), the quotient rule (ln(a/b) = ln a ? ln b), and the power rule (ln(a^b) = b ln a). These identities are central for decomposing logarithms into components with known values.

Approximation Using Known Values

When specific logarithmic values are provided, such as ln 2 and ln 3, these can be used in combination with logarithmic properties to approximate logarithms of other numbers. By expressing numbers as products, quotients, or powers of numbers with known logarithms, the approximation becomes a matter of simple arithmetic based on the log identities.

Transcript

00:01 We need to find the value of natural log of 0 .25.

00:09 So we can write this as equal to natural log of 1 upon 4, which can be written as natural log of 1 minus natural log of 4.

00:19 Since we know that natural log of m upon n is equal to natural log of m minus natural log of n so this will be equal to natural log of 1 minus natural log of 2 square since 2 square is equal to 4 which can be further simplified as natural log of 1 minus 2 twice of natural log of 2 since natural log of x raised to the power of n is equal to n natural log of x now putting the values we are going to get now we know the natural long of 1 is always 0 so this will become 0 minus 2 multiplied with 0 .6931 since the approximate value of log, natural log of n, natural log of 2 is given to us as 0 .6931.

01:14 So putting that we'll get after solving, this is going to be approximately equal to 0 minus 1 .383.

01:31 6 .2 which will be approximately equal to minus 1 .3862.

01:38 Next, we need to find the value of natural log of 24.

01:45 So we can rewrite this as equal to natural log of 3 multiplied with 8, which will be equal to natural log of 3 plus natural log of 8, since we know that natural log of mn is given as natural log of m plus natural log of n from the properties of logarithmic functions.

02:13 So this can be rewritten as natural log of 3 plus natural log of 2 cube since we know that 2 cube is equal to 8.

02:24 So solving this further, we'll get this as equal to natural log of 3 plus 3 times natural log of 2 since we know again from the logarithmic function properties that natural log of x raised to the power of n is equal to n natural log of x now putting the values we are going to get now we are given that log of natural log of 3 is approximately 1 .0986 plus now the value of natural log of 2 given to us is 0 .691 so this will be 3 multiplied with 0 .63931.

03:05 Both of these values are approximate values.

03:08 So the answer will also be approximately 1 .0986 plus 2 .0793 and now adding them both we are going to get the answer approximately equal to 3 .1779.

03:33 Next we need to find the the value of natural log of cube root of 12.

03:43 Now we can rewrite this as natural log of 12 raised to the power of 1 divided by 3 or 1 upon 3.

03:51 Since we know that natural log of x raised to the power of 1 upon n will be equal to 1 upon n multiplied with natural log of x...

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