Use the properties of logarithms and the fact that ln2 ≈0.6931 and ln3 ≈1.0986 to approximate th

Use the properties of logarithms and the fact that ln2...

Key Concepts

Calculator Verification

After obtaining approximations through the application of logarithmic properties and known estimates, using a calculator to verify the results confirms the accuracy of the approximation. This step ensures that any manual simplifications or assumptions made during the calculation process are valid.

Approximation Using Known Logarithm Values

Using known approximate values for certain logarithms, one can estimate the logarithms of other numbers by expressing them in terms of these known values. This is particularly useful when dealing with logarithms that do not have simple exact values, as it allows for a practical and efficient method of approximation.

Properties of Logarithms

This concept covers the fundamental rules that govern logarithms, such as the product, quotient, and power rules. These properties allow complex logarithmic expressions to be simplified into combinations of simpler logarithms, making them easier to calculate or approximate.

Product, Quotient, and Power Rules

These specific rules state that the logarithm of a product is the sum of the logarithms of the individual factors, the logarithm of a quotient is the difference between the logarithms, and the logarithm of a power is the exponent multiplied by the logarithm of the base. These rules are essential tools for breaking down more complicated logarithmic expressions into manageable parts.

Conversion Between Radical and Exponential Forms

Radical expressions can be rewritten as expressions with fractional exponents. This conversion is crucial because it enables the use of the power rule for logarithms, thereby simplifying the process of evaluating logarithms of roots.

Transcript

00:01 All right, on this problem, we're going to use the fact that the natural log of 2 is about 0 .7 or 0 .6 .31, and that the natural log of 3 is approximately 1 ,096.

00:12 Using those facts, we can come up with an approximation for the natural log of 6.

00:17 Now, what this involves is writing 6 somehow in terms of 2 and 3, and then using the properties of log.

00:25 So, what is the natural log of 6? well, that's the same as the natural log of 6.

00:31 Let's see, how about two times three? we're trying to write six in terms of two and three.

00:36 I think that'll do it.

00:37 And now i can expand this product using the properties of logs.

00:42 This becomes the natural log of two plus the natural log of three.

00:46 The log of the product is the sum of the logs.

00:49 So now i can say that that is approximately, when i go from exact to approximate, working straight down, i put the squiggles.

00:56 I get 0 .693 plus 1 .0986.

01:02 And if you want to add those vertically, you're gonna get like 1 .7, 8ish, or something like that.

01:12 Okay, you can add those together.

01:14 Now to verify it on the calculator, let's just see what the natural log of 6 is.

01:19 Boom, 1 .79.

01:22 Actually, if i do this with all the decimals that were given, 0 .6931 plus 1 .0986, i get 1 .79.

01:37 So not 1 .78.

01:38 I get 1 .798.

01:38 I get 1 .71917, which is approximately the exact answer, 1 .7959, so on and so forth.

01:52 So pretty cool.

01:54 Now, we're going to do this multiple times.

01:56 So here's what i'm going to do.

01:57 I'm going to actually save .6931.

02:01 I'm going to store that as alpha a.

02:04 So i can just hit alpha a.

02:05 And i'll put that up here.

02:06 I'll call that a to remind yourself.

02:09 And then up here, i'm going to call this b.

02:11 I'm going to store that 1 .0986.

02:15 Oops.

02:20 1 .0986.

02:21 I'm going to store that as b.

02:23 There we go.

02:25 So what else can we do? well, let's look at another one here, just using those two.

02:30 What would be approximation for the natural log of three halves or 1 .5? well, it's already written in terms of three and two, so we can use the other property of logs.

02:39 It says the log of the quotient is the difference of the logs.

02:42 That becomes the natural log of three minus the natural log of two.

02:46 The factor in the bottom gets its own log in a minus sign.

02:49 So for us, or up here for me, that's like doing b minus a...

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