Understanding Logarithmic Expressions: Calculating log675 with Base 5 × Log3 with Base 3

In this article, we will delve into the process of solving the logarithmic expression (log_{5}675 times log_{3}3). We will break down the problem into simpler steps and demonstrate the calculation process using the change of base formula and the properties of logarithms.

Simplifying log3 with Base 3

First, let's simplify the component of our expression, (log_{3}3). By definition, any logarithm with the same base and argument is equal to 1. Therefore, we have:

(log_{3}3 1)

Now, let's substitute this back into the original expression:

(log_{5}675 times log_{3}3 log_{5}675 times 1 log_{5}675)

Calculating log5 with Base 675 using the Change of Base Formula

To evaluate (log_{5}675), we will use the change of base formula, which is given by:

(log_{a}b frac{log_{c}b}{log_{c}a})

Substituting the problem-specific values into the formula, we obtain:

(log_{5}675 frac{log_{10}675}{log_{10}5})

Breaking Down log10 with Base 675

We can simplify (log_{10}675) by factoring 675:

(675 25 times 27 5^2 times 3^3)

Using the properties of logarithms, specifically that (log a times b log a log b) and (log a^b b log a), we can further break it down:

(log_{10}675 log_{10}(5^2 times 3^3) log_{10}5^2 log_{10}3^3 2log_{10}5 3log_{10}3)

Final Calculation Using Logarithmic Values

Substituting the simplified (log_{10}675) back into the change of base formula, we get:

(log_{5}675 frac{2log_{10}5 3log_{10}3}{log_{10}5})

This expression can be simplified to:

(log_{5}675 2frac{log_{10}5}{log_{10}5} 3frac{log_{10}3}{log_{10}5} 2 3frac{log_{10}3}{log_{10}5})

Given the approximate values for the logarithms:

(log_{10}3 approx 0.4771) (log_{10}5 approx 0.6990)

We can approximate (frac{log_{10}3}{log_{10}5}) as:

(frac{0.4771}{0.6990} approx 0.6826)

Substituting this back into the expression, we get:

(log_{5}675 approx 2 3 times 0.6826 approx 2 2.0478 approx 4.0478)

Therefore, the final value of (log_{5}675 times log_{3}3) is approximately (4.05).

To summarize, the steps involve:

Understanding the definition of logarithms with base 3 and base 5. Applying the change of base formula to transform the expression. Simplifying using logarithmic properties to break down the numbers. Substituting logarithmic values to get a final numerical answer.

Conclusion

Through this detailed explanation, we have successfully calculated (log_{5}675 times log_{3}3) and demonstrated the application of key concepts in logarithmic mathematics. Understanding these techniques can be invaluable for more complex problem-solving in mathematics and data analysis.