In this article, we’ll see the definition of logarithm, explore some of their useful properties, and then get some practice using them. By the end of this post, you’ll be much better prepared for those challenging CAT logarithm problems!

## What is a Logarithm?

**Logarithms** are basically the “opposite” (or *inverse*) of exponentiation. The little “*b*” in the following notation is called the **base** of the logarithm, and we always require *b* > 0 and *b* ≠ 1.

In other words, if you need to know what is the value of log_{2} 512, then what you’re really asking is this: What exponent do I need on 2 to get 512? Since 512 = 2^{9}, now you can say that log_{2} 512 = 9. See how the base of the logarithm (2) becomes the base of the exponentiation?

Of course not every logarithm problem is a straightforward as that. We’ll try our hand at some more challenging questions later.

### Important Bases

There are a couple of bases that are incredibly important: 10 and *e*.

- Logarithm base 10 is called
**common logarithm**. The usual notation for common log is to omit the base completely. log*x*= log_{10}*x*. So if you don’t see a base mentioned, you can safely assume it’s a common log. - We use the base
*e*≈ 2.71828 in the**natural logarithm**. The notation for natural log is “ln.” So ln*x*= log_{e}*x*.

What’s so *natural* about *e*, you might be wondering? Well if you want to know more about this important constant, you might want to peruse this Wikipedia article.

Here are the graphs of common log and natural log alongside log base 2 for comparison.

### Logarithm Properties

Logarithms are useful for a number of reasons, not the least of which is that they have some amazing properties.

Here is a nice refresher for the properties of logarithms: ACT Math Logarithms: What You Need to Know. You could also check out this video to get you started.

In practice, the three most important rules involve logarithms of products, quotients, and powers.

**Product Property:**log_{a}(*x**y*) = log_{a}*x*+ log_{a}*y*.**Quotient Property:**log_{a}(*x*/*y*) = log_{a}*x*– log_{a}*y*.**Power Property:**log_{a}(*x*^{r}) =*r*log_{a}*x*.

Here is a more complete list of logarithm properties.

The logarithm function is also **one-to-one (injective)**, which means that logarithms can be “cancelled” from both sides of any equation.

In other words, if you know that log_{a} *M* = log_{a} *N*, then it must be true that *M* = *N*.

Finally, here are a few more important properties of the logarithm function, *f*(*x*) = log_{a} *x*.

- The domain of
*f*is all positive numbers (*x*> 0). - The range of
*f*is all real numbers (**R**). - The graph of
*f*has a vertical asymptote at*x*= 0. - If
*a*> 1, then*f*is always increasing. If 0 <*a*< 1, then*f*is always decreasing.

## CAT Logarithm Practice

Now let’s practice what we’ve learned!

### Problem 1

Find the largest value of *x* that satisfies log_{4}(3*x*^{2} + 44*x*) = 3.

(A) 3/5 (B) 4/3 (C) 7/4 (D) 8/3

### Problem 2

If log_{a} 2 = 0.3562, log_{a} 3 = 0.5646, and log_{a} 5 = 0.8271, find log_{a} 21600.

(A) 5.129 (B) 4.9873 (C) 4.8661 (D) 3.9012

### Problem 3

If *m* = *n*^{2} = *p*^{3}, then what is log_{m}(*mnp*)?

(A) 1 (B) 11/6 (C) 6 (D) Can’t be determined

### Problem 4

If log(*x* – *y*) – log(*x* + *y*) = log(*y*/*x*), find the value of (*x*/*y*)^{2} + (*y*/*x*)^{2}

(A) 2 (B) 3 (C) 4 (D) 6

### Problem 5

If three positive numbers, *a*, *b*, *c*, are in geometric progression, then which of the following is equivalent to ln *a* + ln *b* + ln *c*?

(A) 3 ln *a* (B) 3 ln *b* (C) 3 ln *c* (D) ln(*a* + *b* + *c*)

## Solutions

### Solution to Problem 1

(B) 4/3

Use the definition of logarithm to rewrite the given equation as an exponential equation.

log_{4}(3*x*^{2} + 44*x*) = 3

3*x*^{2} + 44*x* = 4^{3} = 64.

3*x*^{2} + 44*x* – 64 = 0

(3*x* – 4)(*x* + 16) = 0

There are two solutions, *x* = 4/3 and *x* = -16. Of these two, 4/3 is the largest.

### Solution to Problem 2

(A) 5.129

The key is to break down the number 21600 into its prime factors.

21600 ÷ 2 = 10800

10800 ÷ 2 = 5400

5400 ÷ 2 = 2700

2700 ÷ 2 = 1350

1350 ÷ 2 = 675

So far, we have a factor of 2^{5}.

675 ÷ 3 = 225

225 ÷ 3 = 75

75 ÷ 3 = 25

So the next factor is 3^{3}.

Finally, 25 = 5^{2}. Thus, the prime factorization is: 2^{5}3^{3}5^{2}.

Now use the product property of logarithms….

log_{a} 21600 = log_{a}(2^{5}3^{3}5^{2}) = log_{a}(2^{5}) + log_{a}(3^{3}) + log_{a}(5^{2})

….and the power property:

= 5 log_{a} 2 + 3 log_{a} 3 + 2 log_{a} 5

Finally, we substitute the given information and evaluate.

= 5(0.3562) + 3(0.5646) + 2(0.8271) = 5.129

### Solution to Problem 3

(B) 11/6

Let’s start with the multiplicative property of logarithms.

log_{m}(*mnp*) = log_{m} *m* + log_{m} *n* + log_{m} *p*.

The first term is easy: log_{m} *m* = 1.

For the other terms, we have to use the given equations to rewrite *n* and *p* in terms of *m*.

Since *m* = *n*^{2}, we have: *n* = *m*^{1/2}.

Therefore, log_{m} *n* = log_{m} *m*^{1/2} = 1/2.

In a similar way, since *m* = *p*^{3}, we have: *p* = *m*^{1/3}.

Therefore, log_{m} *p* = log_{m} *m*^{1/3} = 1/3.

In total, we obtain 1 + 1/2 + 1/3 = 11/6.

### Solution to Problem 4

(D) 6

This problem will definitely test our abilities to use the logarithm rules and properties correctly *and* strategically.

First, use the quotient property *in reverse* to combine the logarithms on the left side of the equation. Then you can use the cancellation property to get rid of the logs.

The next few steps involve careful algebraic manipulation. Perhaps a little trial and error is necessary along the way. The key is to get the left side of the equation to match exactly with (*x*/*y*)^{2} + (*y*/*x*)^{2}.

We *almost* have the correct expression on the left side! In order to make it match, square both sides, but be careful with that middle term.

### Solution to Problem 5

(B) 3 ln *b*

**A word of caution:** As tempting as it looks, letter choice (D) is **not** correct! Logarithms do not preserve addition — that is,

log_{a} (*x* + *y*) ≠ log_{a} *x* + log_{a} *y*.

Instead, we have to analyze the given information for clues. Because *a*, *b*, and *c* are in a geometric progression, that means that there is a *common ratio*, r, such that *b* = *ar* and *c* = *ar*^{2}.

Then we can use the product property of logarithms *in reverse*.

ln *a* + ln *b* + ln *c* = ln(*abc*) = ln((*a*)(*ar*)(*ar*^{2})) = ln(*a*^{3}*r*^{3}) = ln((*ar*)^{3})

From this step, use the power property of logarithms to pull the exponent down. Then use the fact that *b* = *ar* to complete the problem.

ln((*ar*)^{3}) = 3 ln(*ar*) = 3 ln *b*

## Summary

Logarithms are inverse to exponentiation. They satisfy many remarkable properties.

Some of the most important properties involve multiplication, division, and exponents:

**Product Property:**log_{a}(*x**y*) = log_{a}*x*+ log_{a}*y*.**Quotient Property:**log_{a}(*x*/*y*) = log_{a}*x*– log_{a}*y*.**Power Property:**log_{a}(*x*^{r}) =*r*log_{a}*x*.

The logarithm function is one-to-one, has domain all positive numbers, and range all real numbers.

With these fundamental rules and properties at hand, you’ll be well-prepared to tackle even the most challenging CAT logarithm questions!

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