Problem 36 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

If \(a=\log _{30} 3\) and \(b=\log _{30} 5\), then find the value of \(\log _{10} 8\)

Step-by-Step Solution

Answer

\(\log _{10} 8 = 1\)

Step 1: Express 30 as product of prime numbers

We can write 30 as the product of prime numbers, that is \( 30 = 2*3*5 \)

Step 2: Express log_10 8 in base 2

We also know that \( 8 = 2^3 \), and hence, \( \log _{10} 8 = \log_{2^3} 2 = 1/3 \)

Step 3: Recalling the logarithmic identities

Using the logarithmic identity \( \log_ab = \log_c a / \log_c b \)

Step 4: Applying the identities

We can now express \( a \) and \( b \) in terms of base 10, as follows: \( a = log_{10} 3 / log_{10} 30, b = log_{10} 5 / log_{10} 30 \). Using the fact that \( log_{10} 30 = log_{10} 3 + log_{10} 2 + log_{10} 5 \), we express \( a \) and \( b \) as follows: \( a = 1/(1+b), b = 1/(1+a) \). Plugging \( b \) into \( a \), we get \( a = 1/(2-a) \).

Step 5: Solving the equation

After rearranging the equation we found in the last step we get \( 2a = 1 + a^2 \). Solving it gives us \( a = 1/2 \) and \( b = 1 \).

Step 6: Find the value of log_10 8

Now, we can substitute the values of \(a\) and \(b\) into the formula \( \log _{10} 8 = 1/(2*a*b) \), to obtain the final result

Key Concepts

Logarithms and Exponential Functions

Understanding logarithms and exponential functions is crucial when dealing with growth patterns or the decay of processes, which frequently appear in various scientific fields. Essentially, a logarithm is the inverse operation to exponentiation. If you have the exponential equation y = b^x, the logarithmic form would be x = log_b(y), indicating how many times you need to multiply the base, b, to get y.

For example, with the exponential function 2^3 = 8, the equivalent logarithmic form is log_2(8) = 3, since you have to raise the number 2 to the power of 3 to get 8. This interchangeable relationship can be a powerful tool when finding unknown variables in either exponents or bases, as demonstrated in the textbook exercise that involves converting an exponential form into a logarithm to solve for the value of \(\log _{10} 8\).

Logarithmic Equations

Logarithmic equations involve variables located within the logarithm, which can sometimes appear daunting to solve. To tackle these, you rely on properties of logarithms to rewrite the equations in a more manageable form. One way to solve such an equation is to convert the log form into an exponential form, which is a direct application of the definition of logarithms.

For instance, if you're given log_b(x) = y, this can be rewritten as the exponential equation x = b^y. In the context of our exercise, rearranging the provided logarithmic expressions in terms of base 10 creates a system of equations that, once solved, will reveal values of a and b that make it possible to calculate \(\log _{10} 8\) using another log identity.

Properties of Logarithms

The properties of logarithms are essential for simplifying complex logarithmic expressions and equations. There are several key properties including:

• The Product Rule: log_b(MN) = log_b(M) + log_b(N)
• The Quotient Rule: log_b(M/N) = log_b(M) - log_b(N)
• The Power Rule: log_b(M^k) = k * log_b(M)
• Change of Base Formula: log_b(M) = log_c(M) / log_c(b)

These properties simplify calculations and help in solving equations where the variable is inside a log function by transforming the relations in ways that reveal the variables.

In our original exercise, using the Change of Base Formula is crucial; it allows us to express a and b using base 10 logarithms, which ultimately helps to solve for the value of \(\log _{10} 8\) by making a connection with previously calculated values.