Why is t= ln(2)/.04 the same

as t= 25ln(2)?
Please explain it step by step. What rule do you use or how do you change the .04 to the 25?

Linda,

Divide 1 by 0.04 using long division.
You will get 25.

think of it as

ln(2) * (1/.4)
and 1/.04= 25

(just use your calculator to find 1/.04 or
1/.04 = 100/4 (multiply top and bottom by 100)
= 25

so ln(2)/.04 = ln(2) * 25
= 25 ln(2)

(I believe Damon answered this same question for you before)

Reiny thank you because I didn't answer Damon's I knew it had to have something to do with 25 being 1/4 of 100 but your explanation of thinking of it as * 1/4 made sense to me so thank you

1/0.04 = 1/(1/25)= 25

1 divided by a number is called the reciprocal of the number.

The reciprocal of a fraction is the upside-down fraction.

To understand why t = ln(2)/0.04 is the same as t = 25ln(2), we need to break down the steps involved.

1. Start with the equation t = ln(2)/0.04.

2. Recall that ln(2) is a logarithm with base e.

3. The property of logarithms states that ln(a/b) is equivalent to ln(a) - ln(b).

4. Applying this property, we can rewrite the equation as t = ln(2) - ln(0.04).

5. Use the property of logarithms again to simplify ln(0.04).

6. Since 0.04 can be expressed as 1/25, we get ln(1/25).

7. Another property of logarithms states that ln(1/x) is equivalent to -ln(x).

8. Using this property, we can rewrite ln(1/25) as -ln(25).

9. Substituting this back into the equation, we have t = ln(2) - (-ln(25)).

10. Simplifying further, we get t = ln(2) + ln(25).

11. The property of logarithms also states that ln(a) + ln(b) is equivalent to ln(a * b).

12. Applying this property, we have t = ln(2 * 25).

13. Simplify the expression within the parentheses, giving us t = ln(50).

14. Finally, we can rewrite ln(50) using the property that ln(e^x) = x.

15. This gives us t = ln(2) + ln(25) = ln(50).

Therefore, we have shown that t = ln(2)/0.04 is the same as t = ln(50), or equivalently, t = 25ln(2).

To summarize, we used the properties of logarithms to simplify the equation step by step, including the properties of subtraction, division, and addition. The .04 was changed to 1/25 using the reciprocal property of fractions, and then further simplified using the property of -ln(x). The final result was obtained by applying the property of multiplication and using the simplification ln(e^x) = x.