00:02
In this question, we are rewriting five different expressions involving the natural logarithm in terms of lawn 5 and long 7.
00:11
To start off, we have lon of 1 over 125, and that number we recognize as 5 to the exponent negative 3.
00:21
And this expression by logarithm laws tells us that, or this logarithm can be rewritten as negative 3 times 1 .5 by logarithm.
00:35
Longer than laws.
00:38
Next, we have lawn of 9 .8, and 9 .8 can be written as 49 over 5, which we can see since 9 .8 is 98 divided by 10.
00:57
And so our expression is long of a power of 7 times a power of 5, which means we can separate the logarithm into a sum of two different algorithms by another exponent law or logarithm law, and the law that tells us that exponents in the anti -logorithm can be written as coefficients, allows us to write this expression in terms of lon 7 and lon 5 like this.
01:40
For the third expression, we have 7 root 7 in the anti -logyithm, which we know can be written as 7 to the exponent 3 over 2, since the square root of 2 is 7 to the 1 half, and exponent laws tell us that when we multiply those two powers of 7, we can add their exponents.
02:03
And that brings us to the final answer of 3 or 2 times 1 .7, since the exponents can be rewritten as coefficients in front of the logarithm.
02:18
Next, we have long of 1 ,225.
02:23
And this, we know can be divided by 5...