The value of $7 \log\left(\frac{16}{15} \right) +5
Question:

The value of 7log(1615)+5log(2524)+3log(8180)7 \log\left(\frac{16}{15} \right) +5 \log\left(\frac{25}{24}\right) + 3 \log\left(\frac{81}{80}\right) is equla to

Updated On: Dec 16, 2024
  • log 2
  • 3
  • 5
  • 7
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The Correct Option is A

Solution and Explanation

7log(1615)+5log(2524)+3log(8180)\therefore 7 \log \left(\frac{16}{15}\right)+5 \log \left(\frac{25}{24}\right)+3 \log \left(\frac{81}{80}\right) =log[(1615)7(2524)5(8180)3]=\log \left[\left(\frac{16}{15}\right)^{7} \cdot\left(\frac{25}{24}\right)^{5} \cdot\left(\frac{81}{80}\right)^{3}\right] =log2=\log 2
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Concepts Used:

Exponential and Logarithmic Functions

Logarithmic Functions:

The inverses of exponential functions are the logarithmic functions. The exponential function is y = ax and its inverse is x = ay. The logarithmic function y = logax is derived as the equivalent to the exponential equation x = ay. y = logax only under the following conditions: x = ay, (where, a > 0, and a≠1). In totality, it is called the logarithmic function with base a.

The domain of a logarithmic function is real numbers greater than 0, and the range is real numbers. The graph of y = logax is symmetrical to the graph of y = ax w.r.t. the line y = x. This relationship is true for any of the exponential functions and their inverse.

Exponential Functions:

Exponential functions have the formation as:

f(x)=bx

where,

b = the base

x = the exponent (or power)