# Difference between revisions of "2020 AMC 12A Problems/Problem 10"

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<cmath>\log_2{n}=8</cmath> | <cmath>\log_2{n}=8</cmath> | ||

− | <cmath>2^{\log_2{n}=2^8</cmath> | + | <cmath>2^{\log_2{n}}=2^8</cmath> |

<cmath>n=256</cmath> | <cmath>n=256</cmath> | ||

Adding the digits together, we have <math>2+5+6=\boxed{\textbf{E) }13}</math> ~quacker88 | Adding the digits together, we have <math>2+5+6=\boxed{\textbf{E) }13}</math> ~quacker88 |

## Revision as of 11:46, 1 February 2020

## Problem

There is a unique positive integer such thatWhat is the sum of the digits of

## Solution

Any logarithm in the form .

so

becomes

Using property of addition, we can expand the parentheses into

Expanding the RHS and simplifying the logs without variables, we have

Subtracting from both sides and adding to both sides gives us

Multiplying by , raising the logs to exponents of base to get rid of the logs and simplifying gives us

Adding the digits together, we have ~quacker88