Problem 36

Question

Given $\ln 4=1.3863$ and $\ln 5=1.6094,$ find each value. Do not use a calculator. $$ \ln (5 e) $$

Step-by-Step Solution

Verified

Answer

The value is 2.6094.

1Step 1: Identify the Expression

The expression we need to solve is $\ln(5e)$. This is asking for the natural logarithm of the product of 5 and the mathematical constant $e$.

2Step 2: Apply Logarithm Product Rule

We use the property of logarithms which states that $\ln(ab) = \ln a + \ln b$. Here, apply this to $\ln(5e)$ which becomes $\ln 5 + \ln e$.

3Step 3: Determine $\ln e$

Remember that $\ln e = 1$, since the natural logarithm of $e$ is always 1 by definition of the natural logarithm.

4Step 4: Substitute and Simplify

Substitute the given values into the transformed expression. $\ln 5 = 1.6094$ and $\ln e = 1$. Thus, $\ln(5e) = \ln 5 + \ln e = 1.6094 + 1$.

5Step 5: Final Calculation

Perform the addition: $1.6094 + 1 = 2.6094$. This is the value of $\ln(5e)$.

Key Concepts

Logarithm PropertiesMathematical Constant eProduct Rule in Logarithms

Logarithm Properties

Logarithms have a set of rules that make calculations easier and help solve complex expressions. One such fundamental property is the **product rule**, which states that for any positive numbers $a$ and $b$, the natural logarithm of their product is the sum of their individual logarithms. This can be written as:

$ \ln(ab) = \ln a + \ln b $.

This rule is incredibly useful in breaking down the natural logarithm of a product into the sum of known logarithmic values. In the exercise given, this allows us to express and solve $ \ln(5e) $ by simplifying it to $ \ln 5 + \ln e $. By learning and applying these properties, complex logarithmic computations become much more manageable. Regular practice will ensure these properties become second nature.

Mathematical Constant e

In logarithms, the continuous growth factor is represented by the mathematical constant $e$, approximately equal to 2.71828. This number is the base of natural logarithms, which are particularly useful for solving equations involving exponential growth or decay. In simpler terms, when you take the natural logarithm of $e$, you get 1. Why? Because $e$ raised to the power of 1 equals $e$ itself, making $ \ln e = 1 $. This property is a direct consequence of how the natural logarithm is defined. This concept is critical when simplifying expressions involving $e$. Knowing this, the solution of a problem such as $ \ln(5e) $ becomes straightforward, as you can directly insert the value of $ \ln e $ being 1 into the equation. Understanding $e$ and its logarithmic connections aids in recognizing growth patterns in scientific fields, finance, and more.

Product Rule in Logarithms

The product rule is one of the cornerstones of logarithmic operations. It simplifies the process of taking the logarithm of a product by transforming it into easier additions. Suppose we need to calculate $ \ln xy $. Instead of calculating it directly, the product rule allows us to break it down:

$ \ln xy = \ln x + \ln y $.

Applying this rule to our exercise for $ \ln(5e) $, we decompose it into $ \ln 5 + \ln e $. This simplifies our work because it allows the use of readily known values such as $ \ln 5 = 1.6094 $ and $ \ln e = 1 $. The ability to effortlessly expand from multiplication to addition means that keeping track of your logarithmic properties will make working with logarithms efficient and intuitive.

Problem 36

Problem 37

Other exercises in this chapter

Problem 36

Differentiate. $$ f(x)=\left(x^{2}+3 x-9\right) e^{x} $$

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Problem 36

Differentiate. $$ g(x)=\sqrt{x^{3}-x}\left(\log _{5} x\right) $$

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Problem 37

Annual consumption of beef per person was about $64.6 \mathrm{lb}$ in 2000 and about $61.2 \mathrm{lb}$ in 2008 . Assuming that $B(t),$ the annual beef co

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Problem 37

Differentiate. $$ f(x)=\frac{e^{x}}{x^{4}} $$

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