Evaluate log base 3 of 6

Problem

(log_3)(6)

Solution

1. Apply the change of base formula to rewrite the logarithm using a more common base, such as the natural logarithm ln() or common logarithm (log_10)()

(log_b)(a)=ln(a)/ln(b)

2. Substitute the values from the expression into the formula where a=6 and b=3

(log_3)(6)=ln(6)/ln(3)

3. Expand the numerator using the product rule for logarithms, ln(x*y)=ln(x)+ln(y) since 6=2×3

(log_3)(6)=ln(2×3)/ln(3)

4. Simplify the expression by splitting the numerator and dividing.

(log_3)(6)=(ln(2)+ln(3))/ln(3)

5. Distribute the denominator to isolate the constant term.

(log_3)(6)=ln(2)/ln(3)+ln(3)/ln(3)

6. Finalize the exact form by simplifying the second term to 1

(log_3)(6)=(log_3)(2)+1

7. Calculate the numerical value using a calculator for an approximate decimal result.

(log_3)(6)≈1.6309

Final Answer

(log_3)(6)=ln(6)/ln(3)≈1.6309

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