In 27-56, evaluate each logarithmic expression. Show all work. (log3 27+8 log16 2)/(log8 512) ·l

step 1 In this exercise, we want to evaluate this logarithmic expression. So all we really have to do i

In 27-56, evaluate each logarithmic expression. Show all...

Key Concepts

Definition of Logarithms

A logarithm represents the exponent to which a base must be raised to produce a given number. It is expressed in the form log_b a = c, meaning that when the base b is raised to the power c, the result is a. This fundamental understanding is essential for interpreting and solving logarithmic equations.

Evaluating Logarithms of Exponential Numbers

When the argument of a logarithm is a power of the base—for example, in the form log_b (b^k)—the logarithm simplifies directly to the exponent k. Recognizing when numbers can be expressed as powers of each other allows for straightforward simplification of logarithmic expressions without resorting to complex calculations.

Properties of Logarithms

Logarithms follow several key properties, including the power rule, product rule, and quotient rule. These properties enable the manipulation and combination of logarithmic expressions, making it possible to simplify and evaluate expressions that involve sums, differences, or multiples of logarithms.

Conversion Between Exponential and Logarithmic Forms

Converting between exponential and logarithmic forms is a powerful technique for evaluating expressions. By rewriting numbers as powers of a common base or using the change-of-base formula, more complex logarithms can be transformed into simpler expressions, facilitating easier calculation and understanding.

Transcript

00:01 In this exercise, we want to evaluate this logarithmic expression.

00:03 So all we really have to do is take each individual part, each individual log that we have, really, and do each one, and then put them all together and we'll get the answer.

00:11 So i think for me, honestly, the one to start with is log base 1000 of 10.

00:15 That's just super simple.

00:16 We can get it out of the way.

00:18 I already know that 10 cubed equals 1 ,000.

00:22 But in this case, we're not looking for log base 10 of 1 ,000.

00:25 We're looking for log base 1 ,000 of 10.

00:27 So knowing 10 cubed of a thousand actually does help us though because we're trying to find a thousand to what power equals 10 that's what we're trying to look for.

00:36 So if we know that 10 cubes is 1 ,000, then we must also know that the cube root of a thousand is going to be 10.

00:42 And if we have a cube root, how do we write that as a power? that would just be 1 third.

00:47 So we know that log based 1 ,000 of 10 is just going to give us 1 third.

00:54 It's 1 ,000 to the 1 3rd power equals 10.

00:56 Now let's start with the denominator that we have here.

00:59 I'll start writing stuff in.

01:01 Where we have log base 8 of 512.

01:03 So to write that as a, in an exponential form, what we can do is we can say 8 to what power, we'll call it x is what we're looking for, equals 512.

01:12 Now let's find a common base for these two sides.

01:16 I know that i can write 8 as 2 cubed, so i'll write the left side as 2 to the 3x power, just replacing 8 with 2 cubed.

01:23 And for 512, i'm assuming that there's also going to be, if i don't already know this memorized, but we can assume that there's going to be a two that divides in.

01:31 And we know that, okay, we know 256 is a power of two.

01:35 So 512 is just double that, so it's also going to be a power of two.

01:38 I know that 128, 64, all of these are powers of two, just doubling them up goes to 512...

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