Logarithm Base

Understanding Logarithm Bases and the Change of Base Formula

Logarithms are a fundamental mathematical concept that is used to solve a wide range of problems in various fields. In this article, we will delve into the topic of logarithm bases and the change of base formula, which is used to solve logarithms with different bases.

Firstly, we will apply the change of base formula on the first term of the expression, resulting in:

loga(x) = logb(x)/logb(a)

Looking at the denominator, we can simplify it either manually or using a calculator. The simplified result will give us the value of 2, as 3 squared is 9. Therefore, our new expression becomes:

loga(x) = 2

Next, we can multiply each term by 2, giving us:

2loga(x) = 4

We can now use the power logarithmic rule on the second expression, which is:

loga(x2)

Applying the rule, we get:

2loga(x) = 4

Using the addition rule, we can combine the two expressions to get:

3loga(x) = 6

To find the value of x, we take the anti-log of both sides:

x = 36

This simplifies to:

x = 729

The change of base formula essentially helps us raise the base a to the power of 6, allowing us to solve logarithms with different bases. This is the final step in solving logarithms with different bases.

When it comes to graphical representations of logarithmic functions, the base of the logarithm can have an impact on the outcome of the graph. Generally, the larger the base, the smaller the curve on the graph. In other words, the larger the base, the closer the curve gets to the y-axis.

Let's take an example to better understand this concept. Say we plot the logarithmic expressions y = log2(x) and y = log3(x). As you can see, log3(x) has a bigger base than log2(x). Plotting the two expressions on a graph and observing the plot (please note that the plot may vary based on the values chosen):

• y = log2(x)
• y = log3(x)

The above graph shows that the curve for log3(x) is closer to the y-axis, indicating the impact of the base on the graph.

Key Points to Remember

Here are some key points to keep in mind when dealing with logarithms and their bases:

• The logarithm base is either the subscript of the logarithm symbol (log) or the number that carries or raises the exponent, depending on the form of the expression.
• The change of base formula (loga(x) = logb(x)/logb(a)) is used to solve logarithms with different bases.
• The bigger the logarithm base, the smaller the curve on the graph. In other words, the larger the base, the closer the curve gets to the y-axis.

How do you find the base of a logarithm?

To find the base of a logarithm, you should be able to identify it. The base is usually the subscript of the logarithm symbol (log). For example, in log2(4), the base is 2.

What is an example of a logarithm base function?

An example of a logarithm base function is y = log2(x).

What are the four laws of logarithm?

The four laws of logarithm are:

• logb(x) + logb(y) = logb(x y) (product law)
• logb(x) - logb(y) = logb(x/y) (division law)
• logb(xn) = n logb(x) (power law)
• logb(x) = loga(x) / loga(b) (change of base law)

How does the base of a logarithm function affect a graph?

The larger the base, the closer the curve gets to the y-axis. The larger the base, the smaller the curve on the graph.

What is the meaning of logarithm base?

The logarithm base is either the subscript of the logarithm symbol (log) or the number that carries or raises the exponent, depending on the form of the expression (by = x or loga(x) = y).