Log Function and Exponent Precedence

In summary, the conversation discusses the interpretation of hand-written log and trig functions compared to entering them into a calculator or Wolfram Alpha. There is a difference in how operators are interpreted, and it is recommended to use parentheses to avoid ambiguity. The conversation also touches on different rules for operator precedence and the importance of writing expressions unambiguously.
  • #1
eurythmistan
5
0

Homework Statement



This isn't really a specific problem, just a question if hand-writing log functions (or trig functions) is interpreted differently than when typing them into a calculator or something like Wolfram Alpha.

Suppose you have this on paper:

ln ex

Is this the same as both of the expressions below?

ln (ex)

ln (e)x

This is what you get, when you enter what I think are equivalent expressions to each of those, onto a calculator (or wolfram)

ln (e^x) ===> x

ln (e^1)^x ===> 1

But I guess my question is, is this really the way you'd interpret the above expressions, if you saw them written out?

I thought that if you wanted the ln function taken to a power, you'd write these, for example:

(ln e)x

lnxe

If you do something similar with sine, then wolfram and my TI-82 calculator differ in their interpretations:

sin(pi/4)^2 =

.5 (according to wolfram)

.5785... (according to ti-82)

I'm wondering if this is a case where there isn't really one specified standard, or if I'm doing something wrong?


Any insight will be appreciated, thank you!
 
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  • #2
The short answer is: avoid ambiguous expressions. Even if you think precedence is well-established, misunderstandings can occur. Use parentheses freely, or just arrange the expression more intelligently.
 
  • #3
eurythmistan said:

Homework Statement



This isn't really a specific problem, just a question if hand-writing log functions (or trig functions) is interpreted differently than when typing them into a calculator or something like Wolfram Alpha.

Suppose you have this on paper:

ln ex

Is this the same as both of the expressions below?

ln (ex)

ln (e)x
I don't see that the two expressions above are different, unless you mean the second to be (ln e)x.
eurythmistan said:
This is what you get, when you enter what I think are equivalent expressions to each of those, onto a calculator (or wolfram)

ln (e^x) ===> x

ln (e^1)^x ===> 1

But I guess my question is, is this really the way you'd interpret the above expressions, if you saw them written out?

I thought that if you wanted the ln function taken to a power, you'd write these, for example:

(ln e)x

lnxe

If you do something similar with sine, then wolfram and my TI-82 calculator differ in their interpretations:

sin(pi/4)^2 =

.5 (according to wolfram)

.5785... (according to ti-82)

I'm wondering if this is a case where there isn't really one specified standard, or if I'm doing something wrong?


Any insight will be appreciated, thank you!
It looks like wolfram and TI have different rules for determining operator precedence. As Curious3141 says, the best thing to do is to write expressions unambiguously so that there will be no confusion. For your trig example, here's what I mean:
[sin(pi/4]]2 vs. sin((pi/4)2).
 
  • #4
Totally makes sense, thanks!

This was actually a question posed to me by someone else, and I tried to say something similar to what you both did, but you both said it so much better. Thanks again!
 

1. What is the difference between a log function and an exponent?

A log function is the inverse of an exponent. It calculates the power to which a base number must be raised to equal a given number. An exponent, on the other hand, is the power to which a base number is raised. For example, in the equation 23 = 8, the exponent is 3 and the base number is 2.

2. How do you solve equations involving both log and exponential functions?

To solve equations involving both log and exponential functions, you can use the property logb(xy) = y*logb(x). This allows you to convert between logarithmic and exponential forms, making it easier to solve for the unknown variable.

3. What is the order of operations for log functions and exponents?

The order of operations for log functions and exponents follows the PEMDAS rule used in algebra. This means that parentheses, exponents, multiplication and division, and addition and subtraction should be performed in that order. However, when there are multiple log or exponential functions, they should be evaluated from left to right.

4. Can you take the log of a negative number?

No, you cannot take the log of a negative number. Logarithms are only defined for positive numbers. If you try to take the log of a negative number, you will get an error. However, you can take the log of a complex number, which can have a negative real or imaginary part.

5. How is log function and exponent precedence used in real-world applications?

The log function and exponent precedence are used in various fields such as finance, engineering, and computer science. In finance, the logarithmic return is used to calculate the growth or decline of an investment over time. In engineering, logarithms are used to convert between different units and to measure the intensity of sound and earthquakes. In computer science, logarithms are used in algorithms to improve efficiency and in cryptography to encode and decode messages.

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