Question

as91261 algebra - homework
homework sheet 2: logarithms
skills: log form; logarithmic rules; solving logarithmic equations
show working. exam-style questions have extra working lines.

1. write \\(2^3 = 8\\) in logarithmic form

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1. write \\(\log_5(125) = 3\\) in index form

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1. evaluate \\(\log(100)\\)

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1. evaluate \\(\log(0.01)\\)

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1. solve \\(\log(x) = 3\\)

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1. solve \\(\log(x) = 0\\)

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1. simplify \\(\log(2) + \log(5)\\)

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1. simplify \\(\log(a) + \log(b)\\)

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1. simplify \\(\log(a) - \log(b)\\)

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1. simplify \\(\log(a^3)\\)

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1. simplify \\(\log(1000) - \log(10)\\)

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1. simplify \\(2\log(3)\\)

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1. solve \\(\log(x) + \log(2) = 3\\)

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1. solve \\(\log(3x) = 2\\)

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1. solve \\(\log(x) = -1\\)

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1. solve \\(\log(x) = 2.5\\)

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1. solve \\(\log(x) - \log(4) = 1\\)

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1. solve \\(\log(x) = \log(7)\\)

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1. solve \\(\log(x) + \log(x) = 2\\)

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1. write \\(\log_3(x) = 4\\) in index form

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1. exam style: simplify \\(\log(50) - \log(2)\\)

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1. exam style: simplify \\(\log(a^{2b})\\)

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1. exam style: simplify \\(\log(a/b^2)\\)

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1. exam style: solve \\(\log(2x) = 1.5\\)

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1. exam style: solve \\(\log(4x) = 2\\)

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1. exam style: if \\(\log(n) = 4.2\\), write \\(n\\) in index form

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1. exam style: write \\(10^n = 1000\\) as a log equation

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1. exam style: a quantity grows by a factor of 10 each step. how many steps from 1 to 1000?

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1. exam style: solve \\(\log(5x) - \log(2) = 1\\)

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1. exam style: state the restriction for \\(\log(x)\\)

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Explanation:

Identify the target question

The user's memory summary indicates a request for "problem 22 involving negative exponents and coefficients" from an exponent worksheet, but the uploaded image is actually a worksheet titled "AS91261 Algebra - Homework Sheet 2: Logarithms". Let's locate Question 22 on this sheet.
Question 22 states: "22. Exam style: Simplify \(\log(a^{2b})\)".
We will solve this specific question using logarithmic rules.

Recall the power rule of logarithms

The power rule of logarithms states that for any positive base and real numbers:

$$ \log_b(x^y) = y \cdot \log_b(x) $$

In this problem, the base is 10 (common logarithm, written as \(\log\)), the argument is \(a^{2b}\), where the base of the exponent is \(a\) and the exponent is \(2b\).

Apply the power rule to the expression

Using the power rule, we can bring the exponent \(2b\) to the front of the logarithm as a coefficient:

$$ \log(a^{2b}) = 2b \cdot \log(a) $$

Write down the final simplified form

The simplified expression is:

$$ 2b\log(a) $$

Answer:

Question 22

\(2b\log(a)\)