Question

Calculate

Just logs

Please add an explanation

Answer 1

The calculation involves solving an exponential operation
`(\(2^2=4\))` and a logarithmic operation(
`\log_210` ≈3.32) using change of base formula). Adding these results gives approximately 7.32.

Sure, let's solve this mathematical expression:
`\(2^2 + \log_2{10}\)`.

The first part of the expression,
`\(2^2\)`, is an example of exponentiation. In this operation, the base number (2) is multiplied by itself for the number of times represented by the exponent (also 2 in this case). So,
`\(2^2 = 4\)`.

The second part,
`\(\log_2{10}\)`, represents a logarithm. A logarithm answers the question: to what exponent must we raise the base to get a specific number? Here, it asks what power you need to raise 2 to obtain 10. We can estimate this value because
`\(2^3 = 8\)` and
`\(2^4 = 16\)`, so the answer lies between 3 and 4.

To calculate it more precisely, we can use a calculator or apply change of base formula which is

`\[\log_b{a} = \frac{\log_c{a}}{\log_c{b}}\]`

where,

`- \(\log_b{a}\)` is the log of 'a' with base 'b',

`- \(\log_c{a}\) & \(\log_c{b}\)` are logs with any common base 'c'.

Applying this formula,

`\[\log_2{10} = \frac{\ln {10}}{\ln {2}} \]` ≈ 3.32193

Adding both parts together gives us

[ 4 + 3.32193 ≈7.32193. ]

So, after evaluating both parts of the expression separately and then combining them, we find that 7.32193 is approximately equal to 7.32 when rounded off to two decimal places.