((1)/(49))^(1+log 7 2)+5^(-log(15)7)

Jasonbradley

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Jun 16, 2024

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Simplifying the Expression: ((1)/(49))^(1+log 7 2)+5^(-log(15)7)

This article will guide you through simplifying the given expression: ((1)/(49))^(1+log 7 2)+5^(-log(15)7). We'll utilize key properties of logarithms and exponents to achieve a simplified form.

Breaking Down the Expression

Let's start by dissecting the expression into manageable parts:

• ((1)/(49))^(1+log 7 2):
  • We can rewrite (1/49) as 7^(-2).
  • Applying the power of a power rule [(a^m)^n = a^(m*n)], we get: 7^(-2(1+log 7 2)).
• 5^(-log(15)7):
  • This part involves a change of base for the logarithm.

Applying Logarithm Properties

1. Change of Base Formula:

* log(a)b = log(c)b / log(c)a, where c is any positive number other than 1.

Applying this to our expression:

• log(15)7 = log(7)7 / log(7)15 = 1 / log(7)15

Now, our expression becomes:

7^(-2(1+log 7 2)) + 5^(-1/log(7)15)

2. Simplifying further:

• 7^(-2(1+log 7 2)): Using the distributive property, we get 7^(-2 - 2log 7 2).

  • Applying the power of a power rule again, we have 7^(-2) * 7^(-2log 7 2).
  • Using the rule a^(-m) = 1/(a^m), we get 1/(7^2) * 1/(7^(2log 7 2)).
  • Simplifying further: 1/49 * 1/(7^(log 7 2^2)).
  • Utilizing the property a^(log a b) = b, we simplify to 1/49 * 1/2^2 = 1/196.

• 5^(-1/log(7)15): Using the rule a^(-m) = 1/(a^m), we get 1/(5^(1/log(7)15)).

Final Solution

Combining the simplified terms, the final simplified expression is:

1/196 + 1/(5^(1/log(7)15))

This is the simplest form of the original expression using the properties of logarithms and exponents.