Question

Express the given logarithm in terms of a and b: if log3=a, log2=b, then log₃18=?

Answer 1

Using the properties of logarithms, if log₃a = a and log₂b = b, then log₃18 = log₃(2 * 3) = log₃2 + log₃3 = b + 1.

Step-by-step explanation:

When confronted with log properties, we know that the logarithm of a product is the sum of the logarithms of its factors. Therefore, log₃18 can be represented as log₃(2 * 3) because 18 is the product of 2 and 3. Applying this property, we can split log₃18 into log₃2 + log₃3.

We've been given log₂b = b and log₃a = a, which means log₂b = log₃b / log₃2. As log₃2 = log₂3 / log₂2 = 1 / b (using the change of base formula), log₃b = b * log₃2. Thus, log₃2 = 1 / b and log₃3 = log₃(3²) = 2 * log₃3, yielding log₃3 = 1 / 2. Therefore, log₃18 = log₃2 + log₃3 = b + 1/2 = b + 0.5.

This derivation showcases the application of logarithmic properties to simplify expressions. By leveraging the properties of logarithms and the given values of log₂b and log₃a, we express log₃18 in terms of b. The step-by-step breakdown demonstrates how to manipulate the logarithmic expression to reach the final answer, which is log₃18 = b + 1/2.

Answer 2

So,
`\( \log_3 18 \)` can be expressed in terms of
`\(a\) and \(b\) as \(2 + b\)`.

To express
`\(\log_3 18\)` in terms of
`\(a\) and \(b\)`, you can use logarithm properties.

The given logarithmic expressions are
`\( \log_3 a = a \) and \( \log_2 b = b \)`.

Now, we want to find
`\( \log_3 18 \)`.

Since 18 can be expressed as
`\(3^2 * 2\)`, we can use logarithm properties to break it down:

`\[ \log_3 18 = \log_3 (3^2 * 2) \]`

By the product rule of logarithms, this can be written as the sum of two logarithms:

`\[ \log_3 18 = \log_3 3^2 + \log_3 2 \]`

Now, apply the power rule, which allows you to bring the exponent down as a coefficient:

`\[ \log_3 18 = 2 \log_3 3 + \log_3 2 \]`

Since
`\( \log_3 3 = 1 \)` (by the definition of logarithms), the expression simplifies to:

`\[ \log_3 18 = 2 + \log_3 2 \]`

Now, substitute
`\(a\) and \(b\)` back in:

`\[ \log_3 18 = 2 + b \]`