Logarithms#1

Algebra Level 3

If $\log_5 M = 3 + \beta$ and $\log_4 N = 2 + \beta$ , where $\beta \in [0,1]$ , which of the following is/are the correct options.

Number of integral values of $M$ is 501
Number of integral values of $N$ is 48
Minimum value of $M+N$ is 141
Minimum value of $|M-N|$ is 61

1 only 2 only 3 and 4 1, 3 and 4 2, 3 and 4

1 solution

Chew-Seong Cheong
Oct 19, 2017

$\begin{cases} \log_5 M = 3 + \beta & \implies M = 5^{3+\beta} & \text{for }\beta \in [0,1] & \implies M \in [5^3,5^4] = [125, 625] \\ \log_4 N = 2 + \beta & \implies N = 4^{2+\beta} & \text{for }\beta \in [0,1] & \implies N \in [4^2, 4^3] = [16, 64] \end{cases}$

Number of integral values of $M$ is $=625-125+1 \color{#3D99F6} = 501 \text{ True}$
Number of integral values of $M$ is $=64-16+1 = 49 \color{#D61F06} \ne 48 \text{ False}$
Minimum value of $M+N$ is $=125+16 \color{#3D99F6} = 141 \text{ True}$
Minimum value of $|M-N|$ is $=125-64 \color{#3D99F6} = 61 \text{ True}$

The correct options are 1, 3, and 4 .

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kent john calago - 3 years, 7 months ago

Logarithms#1

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