((1)/(49))^(1+log72)+5^(-log(15)7)

Jasonbradley

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Jun 16, 2024

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Simplifying the Expression: ((1)/(49))^(1+log72)+5^(-log(15)7)

This article will guide you through simplifying the expression ((1)/(49))^(1+log72)+5^(-log(15)7). We will use logarithmic properties and basic algebraic manipulations to arrive at the final solution.

Understanding the Properties

Before we begin, let's review some key logarithmic properties:

• Logarithm of a power: log_a(b^c) = c * log_a(b)
• Change of base: log_a(b) = log_c(b) / log_c(a)

Step-by-Step Simplification

1. Simplifying the first term:

   • ((1)/(49))^(1+log72) can be rewritten as (7^-2)^(1+log72).
   • Using the power of a power rule, this becomes 7^-2(1+log72) = 7^{(-2 - 2log72)}.
   • Applying the logarithm of a power property, we get 7^{(-2 - log72²)}.
   • Now, we can simplify further: 7^{(-2 - log5184)} = 1/ (7² * 7^log5184) = 1/(49 * 5184) = 1/254016.

2. Simplifying the second term:

   • 5^(-log(15)7) can be rewritten using the logarithm of a power property: 5^(-log(15)7) = (5^(-1))^(log(15)7) = (1/5)^(log(15)7).
   • We can further simplify this using the change of base property: (1/5)^(log(15)7) = (1/5)^(log7/log15).

3. Combining the terms:

   • Now we have the expression 1/254016 + (1/5)^(log7/log15).

4. Final Simplification:

   • While the second term can be simplified further using a calculator, the expression itself cannot be simplified to a single number without using a calculator.

Conclusion

Therefore, the simplified form of the expression ((1)/(49))^(1+log72)+5^(-log(15)7) is 1/254016 + (1/5)^(log7/log15). This solution involves understanding logarithmic properties and applying them to simplify the expression, ultimately arriving at a more manageable form.