How to get rid of log base 10?

To get rid of a log base 10 (also known as a common logarithm), you exponentiate both sides of the equation or inequality with a base of 10. This utilizes the inverse relationship between base-10 logarithms and base-10 exponential functions, effectively canceling out the log₁₀ term.

Understanding Log Base 10

A logarithm with base 10, often written as log(x) without an explicit base, or sometimes log₁₀(x), answers the question: "To what power must 10 be raised to get x?"

For example:

• log(100) = 2 because 10² = 100
• log(1000) = 3 because 10³ = 1000

The process of "getting rid of" a log base 10 is typically needed when you are solving a logarithmic equation or inequality to find the value of the unknown variable.

The Primary Method: Exponentiating with Base 10

The most direct way to eliminate a base-10 logarithm is by applying its inverse operation: exponentiation with base 10.

The Principle of Inverse Operations

Logarithmic and exponential functions with the same base are inverse functions of each other. This means they "undo" each other.

• If you have log₁₀(x), applying 10^(...) will remove the log.
• Specifically, 10^(log₁₀(x)) = x.
• Similarly, log₁₀(10^x) = x.

This fundamental property is key to isolating the variable from within the logarithm.

Step-by-Step Application

To eliminate a log base 10 from an equation or inequality, follow these steps:

1. Isolate the Logarithm: Ensure that the log₁₀ term is by itself on one side of the equation or inequality. If there are other terms (e.g., 2 + log(x) = 5), move them to the other side first (e.g., log(x) = 3).
2. Exponentiate Both Sides: Use each side of the equation or inequality as the exponent for a base of 10.
   • If you have log₁₀(A) = B, it becomes 10^(log₁₀(A)) = 10^B.
   • If you have log₁₀(A) < B, it becomes 10^(log₁₀(A)) < 10^B.
3. Cancel and Simplify: On the side containing the logarithm, 10^(log₁₀(...)) will cancel out, leaving only the argument of the logarithm.
   • 10^(log₁₀(A)) simplifies to A.
   • The other side will remain as 10^B.
4. Solve for the Variable: Proceed to solve the resulting algebraic equation or inequality for the unknown variable.

Practical Example

Let's illustrate with an example similar to the provided information:

Suppose you have the inequality:
log(4x + 3) < 1/3

Here’s how to get rid of the log base 10:

1. Isolate the Logarithm: The log(4x + 3) term is already isolated on the left side.
2. Exponentiate Both Sides with Base 10:
   10^(log(4x + 3)) < 10^(1/3)
3. Cancel and Simplify: The 10^log on the left side cancels out, leaving only the argument.
   4x + 3 < 10^(1/3)
4. Solve for the Variable: Now you can solve this algebraic inequality:
   4x + 3 < 2.154 (approximately, as 10^(1/3) is the cube root of 10)
   4x < 2.154 - 3
4x < -0.846
x < -0.2115

Remember to also consider the domain of the original logarithm: the argument (4x + 3) must be greater than zero.
4x + 3 > 0
4x > -3
x > -3/4 or x > -0.75

Combining both conditions, the solution would be -0.75 < x < -0.2115.

Why This Method Works

This method is effective because of the fundamental definition of logarithms and the concept of inverse functions. Just as subtraction undoes addition, and division undoes multiplication, exponentiation with base 10 undoes a base-10 logarithm.

Function	Inverse Function	Example of Cancellation
Addition (x + a)	Subtraction (x - a)	(x + a) - a = x
Multiplication (a*x)	Division (x / a)	(a*x) / a = x
Log base 10 (log₁₀(x))	Exponentiation base 10 (10^x)	10^(log₁₀(x)) = x or log₁₀(10^x) = x

Important Considerations

• Domain Restrictions: Always remember that the argument of a logarithm must be positive. After solving the equation or inequality, check your solution against the domain of the original logarithmic expression. For log(A), A must be > 0.
• Isolating the Log: This step is crucial. If you have terms multiplied by the logarithm (e.g., 2 * log(x)) or added/subtracted (e.g., log(x) - 1), you must deal with those first before exponentiating. For instance, 2 * log(x) = 6 would become log(x) = 3, then 10^(log(x)) = 10^3.
• Direction of Inequality: When exponentiating both sides of an inequality with a base greater than 1 (like 10), the direction of the inequality sign remains the same. If the base were between 0 and 1, the inequality direction would flip, but this is not applicable for base 10.

By understanding and applying the inverse relationship with base 10 exponentiation, you can effectively "get rid of" a log base 10 and solve logarithmic expressions.