Logarithms are the inverse operations of exponentials. If a to the power of y equals x, then y is the logarithm base a of x. Today we'll learn how to rewrite logarithms in different bases, specifically converting to natural logarithms. Our main problem is to rewrite log base 3 of 20 using natural logarithms.
To convert between different logarithm bases, we use the change of base formula. Starting with y equals log base a of x, we convert to exponential form a to the y equals x. Taking log base b of both sides gives us log base b of a to the y equals log base b of x. Using the power rule of logarithms, this becomes y times log base b of a equals log base b of x. Solving for y gives us y equals log base b of x divided by log base b of a. Substituting back our original definition of y, we get the change of base formula: log base a of x equals log base b of x divided by log base b of a. This formula works for any valid base b.
Natural logarithms are logarithms with base e, where e is approximately 2.718. We denote natural logarithms as ln of x, which is equivalent to log base e of x. Natural logarithms are fundamental in mathematics and science because of their special properties in calculus. When we apply the change of base formula with base b equal to e, we get log base a of x equals natural log of x divided by natural log of a. This is particularly useful because most calculators have a built-in natural logarithm function.
Now let's apply the change of base formula to our main problem: rewriting log base 3 of 20 using natural logarithms. In our formula, a equals 3 and x equals 20. Substituting these values, we get log base 3 of 20 equals natural log of 20 divided by natural log of 3. To verify this makes sense, we note that both natural log of 20 and natural log of 3 are positive real numbers, so their ratio is meaningful. This gives us our final answer: log base 3 of 20 equals natural log of 20 divided by natural log of 3.
Let's verify our answer numerically. If we've correctly converted log base 3 of 20 to natural logarithms, then 3 raised to the power of our answer should approximately equal 20. Calculating 3 to the power of 2.727, we get approximately 20. This confirms that our conversion using the change of base formula is correct. The small difference is due to rounding in our decimal approximation, but it's close enough to validate our method.
Let's practice with a couple more examples to reinforce our understanding. First, to convert log base 2 of 50 to natural logarithms, we apply the formula: log base 2 of 50 equals natural log of 50 divided by natural log of 2. Numerically, this is approximately 3.912 divided by 0.693, which equals about 5.644. Second, for log base 5 of 100, we get natural log of 100 divided by natural log of 5. This evaluates to approximately 4.605 divided by 1.609, which is about 2.862. These examples demonstrate the general pattern: to convert any logarithm to natural logarithms, we always use the same formula: log base a of x equals natural log of x divided by natural log of a.