Log 10^-7 = x?

Question

Log 10^-7 = x.?
Does x = 1 or
Does x = -7 ?
Or something else?

Answer 1

You can re-write this as:
(-7)(log 10) = x
The base of a logarithm is 10, unless stated otherwise. Since log {subscript 10} 10 is equal to 1, you get:
(-7)(1) = x
So, x = -7

Answer 2

The "^" sign means "to the power of," so the question is asking what is the log of 10 to the power of minus 7, which is log 1/(10^7) or log 1/10,000,000 = log(0.0000001) = -7.

By the way, I understand your confusion. This equation is ambiguous and should have been written with parenthesis. Since precedence is not usually defined for the "log" function (see links below), you could legitimately interpret the left expression as any of:

1. (Log 10)^(-7) = 1

2. Log (10^(-7)) = -7

3. Log base 10 (^-7) = meaningless expression

In the absence of established convention, we might guess that the author meant to group "10^-7" because there are no spaces in the group, yet there is a space after "Log," suggesting that #2 is correct, and because it is common (though technically incorrect) to casually write the argument of a transcendental without parenthesis. Also, the computation of version #1 is not very interesting and seems unlikely to appear in a math book or tutorial.

If the expression as written above is your interpretation of the actual problem and the problem as stated showed the "-7" written as a superscript to "10," then #2 is most likely correct, since there is an implied grouping of a number or variable and its exponent. This is still sloppy notation, though, and should have been written with parenthesis.

One argument for #1 (x = 1) is that computation generally proceeds from left to right unless otherwise dictated by operator precedence. This is a weak argument, though, since there are clearly missing parenthesis somewhere, so I might vote for this option only if there were a space after, but not before, the "10."

I only mention #3 because it's very common to write "Log10" as meaning log in base ten. I reject that formulation offhand, though, because it results in a meaningless expression. In case I've not already been critical enough of the problem, the author should have specified the base. I've simply guessed that the base is 10 because that gives us a nice round answer, but in fact, unless otherwise stated, mathematicians and computers usually assume that "log" with no base shown is in base e.

If you're forced to choose, I would bet on #2 (x = -7), but if it's an essay question, I would assert that the problem cannot be unambiguously parsed and has two correct answers (1 and -7).

Answer 3

ok so when you have a log like this one that does not have a subsrcipt number (a small number at the bottom of log, the opposite of an exponent.) the subscript = 10. So from here on when I put _, the number following is a subscript.
When you have the genral form of x=log_b y. and y > 0, b > 0, and b cannot equal 1... y = b^x
so your equation can be written as
Log_10 10^-7 = x, where 10=b and 10^-7 =y
10^-7 = 10^x
Now since you hav the same coeficent (10) on each side of the equation you can set the powers equal to each other...
-7 = x
or
x = -7

Answer 4

x=-7

Answer 5

(-7) log 10 = log x
(-7) (1) = log x
(-7) = log x
x = 10^(-7)

Answer 6

-7 (Im not 100%)

Answer 7

log10^(--7) = x => x = --7.

Answer 8

u know!