Question

Assume \(\log _{b} x=0.36, \log _{b} y=0.56\), and \(\log _{b} z=0.83 .\) Evaluate the following expressions. $$\log _{b} \frac{b^{2} x^{5 / 2}}{\sqrt{y}}$$

Short answer

Question: Given that \(\log_b{x} = 0.36\), \(\log_b{y} = 0.56\), and \(\log_b{z} = 0.83\), calculate the value of \(\log_b{\frac{b^2 x^{5/2}}{\sqrt{y}}}\). Answer: The value of the given expression is \(\log_b{\frac{b^2 x^{5/2}}{\sqrt{y}}} = 2.62\).

Step-by-step solution

Simplify the expression within the logarithm

First, we will simplify the expression within the logarithm: $$\frac{b^2 x^{5/2}}{\sqrt{y}} = b^2 \cdot x^{5/2} \cdot y^{-1/2}$$

Apply the logarithm to the expression

Now, we will apply the logarithm: $$\log_b{b^2 \cdot x^{5/2} \cdot y^{-1/2}}$$ Using the logarithmic product rule, which states that \(\log_b{AB} = \log_b{A}+\log_b{B}\), we can simplify this expression as: $$\log_b{b^2} + \log_b{x^{5/2}} + \log_b{y^{-1/2}}$$

Use the logarithmic power rule

Now we will use the logarithmic power rule, which states \(\log_b{A^p} = p \cdot \log_b{A}\). Applying this rule to our expression, we get: $$2\log_b{b} + \frac{5}{2}\log_b{x} - \frac{1}{2}\log_b{y}$$

Substitute the given values

We are given \(\log_b{x}=0.36\), \(\log_b{y}=0.56\). Also, we know that for any base \(b\), \(\log_b{b} = 1\). Using this information, substitute the values in the expression: $$2\cdot1 + \frac{5}{2}\cdot0.36 - \frac{1}{2}\cdot0.56$$

Calculate the result

Now, calculate the result of the expression: $$2 + \frac{5}{2}\cdot0.36 - \frac{1}{2}\cdot0.56 = 2 + 0.9 - 0.28 = 2.62$$ So, the value of the given expression is: $$\log_b{\frac{b^2 x^{5/2}}{\sqrt{y}}} = 2.62$$

Key concepts

Logarithmic Rules

Logarithmic rules are the foundation of working with logarithmic expressions. There are several important rules that help to simplify complex logarithmic problems.

• Product Rule: The product rule states that \(\log_b{(A \times B)} = \log_b{A} + \log_b{B}\). This rule is useful when you are multiplying two numbers inside a logarithm.
• Quotient Rule: This rule allows you to simplify division within a logarithm: \(\log_b{(A/B)} = \log_b{A} - \log_b{B}\). It is essential when dividing numbers inside a logarithm.
• Power Rule: According to this rule, \(\log_b{A^p} = p \cdot \log_b{A}\). It becomes handy when dealing with exponents inside a logarithm. Exponents can be moved in front of the logarithm, making calculations more manageable.

Applying these rules can turn a complicated logarithmic expression into something straightforward. For example, simplifying \(\log_b{b^2 \cdot x^{5/2} \cdot y^{-1/2}}\) involves using the product and power rules efficiently to break it down into a sum and difference of simpler logarithmic terms.

Logarithmic Identities

Logarithmic identities are special equations that hold true for all permissible values oftentimes used to simplify expressions or solve logarithmic equations.

• Identity of Base: An important identity is \(\log_b{b} = 1\), which tells us that the logarithm of a base to the power of one is always 1. This is helpful in simplifying terms such as \(\log_b{b^2}\), as it simplifies to \(2 \cdot \log_b{b} = 2\).
• Change of Base Formula: Although not directly used in the exercise, this identity \(\log_b{A} = \frac{\log_c{A}}{\log_c{b}}\) is critical for converting logs from one base to another.

These identities not only help simplify the expressions but also make calculations much simpler. They ensure that even without exhaustive calculations, you can rely on these to break down and solve different logarithmic problems.

Logarithmic Expressions

Logarithmic expressions involve the manipulation of logarithms to express values in an alternative form. They serve as a powerful tool in solving equations and interpreting scientific data.

• Representation: A logarithmic expression typically includes a base and an argument, like \(\log_b{x}\). Here, \(b\) is the base, and \(x\) is the argument.
• Simplification: Using logarithmic rules and identities, complex expressions can be simplified. For example, during expression evaluation, simplify \(\log_b{\frac{b^2 x^{5/2}}{\sqrt{y}}}\) by breaking it down into simpler parts using rules like the product and power rules.
• Substitution: When given specific values like \(\log_b{x} = 0.36\), these can be substituted into partially simplified expressions to compute a numerical result.

Such expressions are routinely manipulated in mathematical problems like the given exercise, transforming complex ideas into numerical results through strategic application of logarithmic principles and the given values.