Question

Evaluate the logarithm. \(\log _7 49\)

Short answer

2

Step-by-step solution

Understand the question

We are asked to evaluate \(\log _7 49\). That means we need to find what power (or exponent) 7 needs to be raised to get 49.

Use the definition of a logarithm

To answer this question, we can use the definition of a logarithm: if \(a^b = c\), then \(\log _a c = b\). Therefore, \(\log _7 49\) asks us what power 7 must be raised to get 49.

Identify the power

In this case, we know that \(7^2 = 49\). Therefore, 2 is the power we need.

Write the answer

So, \(\log _7 49 = 2\).

Key concepts

Exponents

Exponents are a way to express repeated multiplication of the same number. When you see a number written as a base with an exponent, such as \(a^b\), it means that the base \(a\) is being multiplied by itself \(b\) times. For example, \(3^2\) means \(3 \times 3\), which equals 9.
Exponents are a fundamental concept in math, as they simplify expressions and calculations.
They turn long multiplication processes into more manageable forms.

• An exponent is also referred to as a "power."
• \(a^1\) is simply \(a\), since any number to the power of 1 is itself.
• \(a^0 = 1\) for any non-zero number \(a\), which is a useful fact in many calculations.

Base

The base in an exponential expression is the number that is repeatedly multiplied. It's the number under the exponent we keep working with. For example, in \(7^2\), the base is 7.
In the context of logarithms, the base is equally crucial.
A logarithm always has a base, which is typically written as a subscript, like \(\log_7\) in the exercise.

• The base indicates the number you're comparing powers of.
• Different bases, like 2 or 10, allow us to explore various logarithmic forms and scales.

Understanding the base is key to mastering both exponents and logarithms, as it helps determine the structure of the equation you are working with.

Evaluate Logarithm

Evaluating logarithms is about finding the power needed for a base number to achieve a specified result. In the exercise \(\log _7 49\), we want to know "To what power should we raise 7 to get 49?"
This is at the heart of logarithms - they ask the reverse of what exponents do.
To evaluate the log, you should:

• Identify the base number and what result you want.
• Find which power the base needs to reach that result using multiplication.
• Write out the answer clearly.

In our exercise, since \(7^2 = 49\), we quickly see that the answer is 2. Evaluating logs can often require practice with exponents to become intuitive.

Definition of Logarithm

A logarithm is essentially a question about exponents.
The notation \(\log_a b = c\) means that you want to know the power \(c\) to which the base \(a\) needs to be raised to result in \(b\). It's the inverse operation of exponentiation.

• This definition helps solve various types of problems where you want to "reverse engineer" exponents.
• You can use properties and rules like the Product Rule, Quotient Rule, and Power Rule to simplify complex logarithmic expressions.

Understanding logarithms is crucial for higher-level mathematics, especially in solving exponential equations and analyzing growth patterns in sciences.
They make handling very large or very small numbers more manageable.