# Evaluate Logarithms With Natural Base Worksheet

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Evaluating logarithms with the natural base $$e$$ involves finding the exponent that $$e$$ must be raised to equal a given number. The natural logarithm, denoted as $$\ln x$$, is the inverse of the exponential function $$e^x$$. This means $$\ln(e^x) = x$$ and $$e^{\ln x} = x$$. For example, $$\ln(e^3) = 3$$ because $$e$$ raised to the power of $$3$$ equals $$e^3$$. Use this worksheet to enhance your understanding on logarithms.

Algebra 2
Logarithms

## How Will This Worksheet on "Evaluate Logarithms with Natural Base" Benefit Your Student's Learning?

• Helps with calculus, like finding slopes and areas under curves using exponential and logarithmic functions.
• Used in science and economics to predict how things grow or decrease quickly.
• Improves problem-solving by dealing with equations that have logarithms and exponents.
• Gives exact ways to figure out how things change and transform.
• Useful in finance for figuring out how investments grow over time.

## How to Evaluate Logarithms with Natural Base?

• The natural logarithm $$\ln x$$ with base $$e$$ is the exponent to which $$e$$ must be raised to equal $$x$$.
• $$\ln x$$ and $$e^x$$ are inverse functions, meaning $$\ln(e^x) = x$$ and $$e^{\ln x} = x$$.
• To evaluate $$\ln(e^3)$$, recognize that $$\ln(e^3) = 3$$ because $$e$$ raised to the power of $$3$$ equals $$e^3$$.

## Solved Example

Q. Evaluate. Write your answer as a whole number, proper fraction, or improper fraction in simplest form.$\newline$$\frac{\ln (e)}{10} =$ ______
Solution:
1. Evaluate expression: We need to evaluate the expression $\frac{\ln(e)}{10}$. The natural logarithm of $e$, denoted as $\ln (e)$, is a special value in mathematics. The natural logarithm function is the inverse of the exponential function, so $\ln(e)$ is asking for the power to which $e$ must be raised to get $e$. Since $e$ to the power of $1$ is $e$, $\ln(e)$ equals $1$.
2. Substitute $\ln(e)$ value: Now that we know $\ln(e)$ equals $1$, we can substitute this value into our original expression.$\newline$ So, $\frac{\ln(e)}{10}$ becomes $\frac{1}{10}$.
3. Simplify fraction: The fraction $\frac{1}{10}$ is already in its simplest form. There is no common factor between the numerator and the denominator other than $1$, so it cannot be simplified further.

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