Problem 32
Question
In Exercises \(31-46,\) write each expression as a logarithm of a single quantity and then simplify if possible. Assume that each variable expression is defined for appropriate values of the variable(s). Do not use a calculator. $$\log 4.1+\log 3$$
Step-by-Step Solution
Verified Answer
The sum of the logarithms \( \log {4.1}+ \log {3} \) is equivalent to the single logarithm \( \log {12.3} \)
1Step 1: Identify that the logarithms have the same base
Here, both logarithms are base 10 logarithms which is the common or decimal logarithm.
2Step 2: Apply the logarithm's sum-product property
By the sum-to-product rule, adding two logarithms together turns into multiplication inside a single logarithm. So, \( \log {4.1}+ \log {3}= \log {(4.1 \times 3)} \)
3Step 3: Simplify inside the logarithm
Now, we simplify the multiplication inside the logarithm: \( \log {(4.1 \times 3)} = \log {12.3} \)
Key Concepts
Sum-Product PropertySimplifying Logarithmic ExpressionsBase 10 Logarithms
Sum-Product Property
The sum-product property is a key technique when working with logarithmic expressions. It transforms the addition of two logarithms into the logarithm of a multiplied product. The specific rule for this property states that \( \log_b(a) + \log_b(b) = \log_b(a \times b) \). This property only works when the bases of the logarithms are identical.
In the given exercise, we had both \( \log(4.1) \) and \( \log(3) \) with a common base 10, often referred to as base 10 logarithms or common logarithms. The sum-product property allowed us to combine these into a single expression: \( \log(4.1 \times 3) \).
Understanding this property is crucial because it simplifies complex logarithmic equations, turning sums into a single logarithmic expression which is often easier to handle.
In the given exercise, we had both \( \log(4.1) \) and \( \log(3) \) with a common base 10, often referred to as base 10 logarithms or common logarithms. The sum-product property allowed us to combine these into a single expression: \( \log(4.1 \times 3) \).
Understanding this property is crucial because it simplifies complex logarithmic equations, turning sums into a single logarithmic expression which is often easier to handle.
Simplifying Logarithmic Expressions
Simplifying logarithmic expressions involves using properties of logarithms to reduce them to their simplest form. In this problem, after using the sum-product property, the expression \( \log(4.1 \times 3) \) became \( \log(12.3) \).
The goal is to reduce the expression to a form that cannot be further simplified without a calculator. Here, the multiplication results in a single number, making the logarithm simple and concise.
Key steps in simplifying include:
The goal is to reduce the expression to a form that cannot be further simplified without a calculator. Here, the multiplication results in a single number, making the logarithm simple and concise.
Key steps in simplifying include:
- Applying logarithmic properties to combine or break apart expressions.
- Calculating any arithmetic in the argument if possible.
- Leaving the result as a single logarithmic term when further simplification isn’t feasible.
Base 10 Logarithms
Base 10 logarithms, also known as common logarithms, are an important type of logarithm often used in calculations and mathematical problems. They are written as \( \log(x) \) without explicitly showing the base, which is implicitly 10. These are frequently used because they align with the decimal number system, making them convenient for manual calculations.
In our exercise, both logarithmic terms involved base 10: \( \log(4.1) \) and \( \log(3) \). This is consistent with the common logarithm's standard: the base is understood to be 10 unless otherwise specified.
In our exercise, both logarithmic terms involved base 10: \( \log(4.1) \) and \( \log(3) \). This is consistent with the common logarithm's standard: the base is understood to be 10 unless otherwise specified.
- The simplification processes using these logarithms do not require calculator use.
- They are a default in calculations unless another base is specified, such as natural logarithms with base \( e \).
- Base 10 logs are represented simply as \( \log \) rather than \( \log_{10} \).
Other exercises in this chapter
Problem 32
Solve the exponential equation. Round to three decimal places, when needed. $$4 e^{x}=-x+3$$
View solution Problem 32
State whether each function is one-to-one. $$f(x)=-3 x^{2}+1$$
View solution Problem 32
Evaluate each expression without using a calculator. $$\log _{1 / 3} 9$$
View solution Problem 32
Sketch the graph of each function. $$g(x)=2^{3 x}$$
View solution