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

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

Key Concepts

Properties of Logarithms

This concept covers the fundamental rules of logarithms such as 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 properties are essential for breaking down and rewriting complex logarithmic expressions into simpler parts or expressing them in terms of other known logarithms.

Exponentiation and Radical Conversion

This involves understanding how exponents and radicals relate, such as rewriting numbers like 125 as 5³ or converting square roots into exponent form by recognizing that ?a equals a^(1/2). This conversion is critical when expressing logarithms in terms of ln 5 and ln 7, as it simplifies the expression to a more manageable form.

Factorization and Expression Rewriting

This concept entails decomposing numbers into factors that highlight the involvement of the desired base numbers. For instance, identifying that 1225 can be written as 35² (and further breaking down 35 into factors of 5 and 7) allows one to express complex logarithms solely in terms of ln 5 and ln 7. Factorization helps in aligning the given logarithm with the target form by isolating the components that match the known logarithms.

Algebraic Manipulation of Logarithmic Expressions

This includes the strategic use of algebraic operations to combine or separate logarithmic expressions, as well as simplifying fractions and compound expressions. Such manipulation is necessary when re-expressing the logarithms to achieve an expression solely in the terms of ln 5 and ln 7, ensuring that the relationships among the original factors are maintained through precise calculation.

Transcript

00:01 This question is asking us to express the falling logarithms in terms of natural log of 5 and natural log of 7.

00:08 So for the first one, we're given natural log of 1 over 125.

00:19 So this question is asking us to use theorem 2.

00:23 So there's four different ones there.

00:25 This one looks just like the reciprocal rule.

00:28 I see i have a fraction of 1 over 125.

00:33 And using that rule, it told us that this is just going to equal the negative, the natural log of whatever's in the denominator.

00:42 So here that's 125.

00:45 So i want this to be in terms of natural log of 5 or natural log of 7.

00:50 So i'm going to need to manipulate this 125 in order to get a 5 or a 7.

00:55 Well, i know that 5 definitely goes into it.

00:58 And in fact, i know that 125 is actually just equal to.

01:03 5 cubed.

01:09 So we're really close.

01:10 We just want this to be a 5 in here though.

01:11 We have this cubed with it, but we also know from our power rule that we can move our power over to the front.

01:21 So if i do that, i end up getting that this is equal to negative 3 times the natural log of 5.

01:34 In part b, we are given the natural log of 9 .8.

01:41 So for this one i see i do have a decimal on here.

01:45 I know for that division rule i can use that but has to be in fraction form.

01:50 So my first step when i see this is i want to change this into a fraction.

01:56 So i know that 9 .8 is just going to equal 98 over 10.

02:03 I want this to be in terms of fives and sevens though and for that i want this to be reduced as much as possible.

02:10 So if i divided two on top of bottom i get 49 over 5.

02:17 So this is going to equal a natural log, 49 over 5 this can't be reduced anymore so using my division rule i'm going to go ahead and separate this into the natural log of 49 minus natural log of 5 so i already have one part that's in that natural log of 5 or 7 have this natural log of 5 i need this one to change though however i do know that 49 is just actually equal to 7 squared and then again using my power rule i can move that exponent out into the front so i get 2 times natural log of 7 minus the natural log of 5.

03:05 In part c we were given the natural log of 7 times square root of 7.

03:13 This is a product.

03:14 I know i already want that 7 to be there, so i want to go ahead and separate this into the natural log of 7 plus the natural log of square root of 7 using my product rule.

03:25 I know for a square root, that's just the same thing as this being 7 to the 1 half...

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