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Write c in terms of $\log_{10} A$, $\log_{10} B$, and $\log_{10}(c + 1)$ - Leaving Cert Mathematics - Question e - 2022

Question e

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Write c in terms of $\log_{10} A$, $\log_{10} B$, and $\log_{10}(c + 1)$. A student got 80% on a guitar exam. After two years of not playing the guitar, the stude... show full transcript

Worked Solution & Example Answer:Write c in terms of $\log_{10} A$, $\log_{10} B$, and $\log_{10}(c + 1)$ - Leaving Cert Mathematics - Question e - 2022

Step 1

Write c in terms of $\log_{10} A$, $\log_{10} B$, and $\log_{10}(c + 1)$

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Answer

From the model equation, we have:

$A = B(c + 1)$

Taking logarithm base 10 on both sides gives:

$\log_{10} A = \log_{10}(B(c + 1))$

Using the logarithm property:

$\log_{10} A = \log_{10} B + \log_{10}(c + 1)$

Rearranging gives:

$c = \frac{A}{B} - 1$

Step 2

A student got 80% on a guitar exam. After two years of not playing the guitar, the student got 47% on the same exam. Use this to find the value of c in the model above, correct to 3 decimal places.

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104 rated

Answer

Let:

$A = 80$ (initial skill level)
$B = 47$ (skill level after 2 years)

We can use the formula derived:

$c = \frac{A}{B} - 1$

Substituting the values gives:

$c = \frac{80}{47} - 1$

Calculating this:

$c = 1.702 - 1 = 0.702$

Hence, the value of $c$ is approximately (0.702) when rounded to three decimal places.

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