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Solving exponential equations using logarithms: base-2

We can use logarithms to solve *any* exponential equation of the form a⋅bᶜˣ=d. For example, this is how you can solve 3⋅10²ˣ=7:
1. Divide by 3: 10²ˣ=7/3
2. Use the definition of logarithm: 2x=log(7/3)
3. Divide by 2: x=log(7/3)/2 Now you can use a calculator to find the solution of the equation as a rounded decimal number
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Created by Sal Khan.

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  • leaf green style avatar for user Frontiers
    Is this the first video that introduces log or have I missed something? I'm working through the content in order and have found that this plus some of the previous exercises have required knowledge of the ln and e concepts which I don't have.
    (40 votes)
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  • blobby green style avatar for user dugee23
    i dont understand how log(base)2^2t = t ?
    (14 votes)
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    • piceratops tree style avatar for user Theresa Johnson
      First, I think you must have typed the problem wrong - this only makes sense as log (base 2) 2^t with the t as the exponent. If this is not the case - let me know and I'll try to figure out what's happening.

      Basically a log is the answer to what power of the given base gives the expression. So log (base 2) 2^t means

      2 to what power equals 2^t. Since the bases match, you just need to name the power.
      (24 votes)
  • blobby green style avatar for user Vivian Leung
    I'm just curious: is there a way to calculate logarithms without your calculator, and if not, why?
    I'm just wondering how mathematicians back in the day would have solved this problem since they don't have a calculator to do everything for them.
    (17 votes)
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    • marcimus pink style avatar for user PapaJoe
      Actually, most text books on algebra and trigonometry years ago would have tables in the back for common functions like log, ln, sin, cos, tan, etc. so that a student would look up the values in a table and interpolate between the 2 values that were close.
      (11 votes)
  • blobby green style avatar for user Keith Whittingham
    It seems to me to be easier and more logical to do the following...

    5 • 2^t = 1111
    => 2^t = 1111 / 5 // Divide by 5 to isolate the exponential term
    => ln(2^t) = ln(1111 / 5) // Take logs of both sides log or ln don't care
    => t • ln(2) = ln(1111 / 5) // Log magic! Exponent inside same-same multiply outside
    => t = ln(1111 / 5) / ln(2) // Divide by ln(2)
    => t = 7.7957... // Squeezed out of a calculator
    (15 votes)
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  • blobby green style avatar for user leyyaa.bamath
    why do you use log base 2
    (12 votes)
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  • blobby green style avatar for user Micah Larroque
    I am just curious about a more detailed exponential equation:
    4^(5x-x^2)=4^-6
    (4 votes)
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    • blobby green style avatar for user Pacho Cambiaso
      First, you have to know that the base on both sides of the equation is 4, so now you have to add the log base 4 of both sides of the equation:

      log [base4] 4^(5x-x^2) = log [base 4] 4^-6

      Why do you do this? because one of the properties of logarithms says that
      log [base a] a^x = (just as a^(log [base a] x) = x)

      so now canceling the 4's with the log [base4] you get that 5x-x^2 = -6

      next, by adding 6 to both sides of the equation you get -x^2 + 5x + 6 = 0

      by multiplying by negative one (so you have x^2 instead of -x^2) you get:

      x^2-5x-6=0

      by factoring you get:

      (x-6) (x+1) = 0

      so X1=6 X2= -1
      (3 votes)
  • blobby green style avatar for user ieltsbestmytest1
    I am trying to solve the same problem using a different approach but I do not get the same answer as Sal. My approach is
    y = 5.2^t
    5.2^t=1111 // since we want to find for 1111
    // now from here i go on a different route than Sal
    log (base 5.2) 1111 = t
    log 1111 / log 5.2 = t
    // after using the calculator I am getting below value for t
    t=4.254

    Why is my answer different? Am I doing something wrong?
    (3 votes)
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  • blobby green style avatar for user michelle.endozo
    How do I make the bases the same in an exponential equation?
    (1/9)^-3x =3^3
    (3 votes)
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  • blobby green style avatar for user mitchill24
    How would I solve this equation?

    16^-2x * 64^-2x = 64
    (2 votes)
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    • stelly blue style avatar for user Kim Seidel
      Change to a common base: 16 = 4^2 and 64 = 4^3
      So, your expression can be changed into:
      (4^2)^(-2x) * (4^3)^(-2x) = 4^3
      Simplify the exponents on the left. One exponent raised to another means we multiply the exponents.
      4^(-4x) * 4^(-6x) = 4^3
      Simplify further. We have a common base on the left and the 4's are being multiplied, so add their exponents.
      4^(-10x) = 4^3
      Now, the only way for these 2 sides to be equal are it the exponents are equal. So, solve: -10x=3 to find x.

      Hope this helps.
      (3 votes)
  • duskpin ultimate style avatar for user Izzy
    For what use would I need a log base 2? I gave it a go and came up with 1111/5=2^t followed by log base 2 (1111/5)=t ........ then Log(1111/5) divided by Log(2)=t .....Yay same answer.... so why do I need to use a base of two? I'm confused
    (3 votes)
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Video transcript

Voiceover:Let's say that we've got the function y is equal to five times two to the t power. Someone were to come up to you and say, "Hey look, this is an interesting function." "But I'm curious, I like the number 1,111." "I'm curious at what point for what t value" "will my y be equal to 1,111." I encourage you to pause this video and think about it on your own. At what t value will this y be equal to a roughly equal to 1,111. If you see the need you might want to use a calculator. I'm assuming you've given a go at it. Let's work through this together, so we want to say, when does five times two to the t power, equal 1,111. Let's write that down. When does five times two to the t power equal 1,111. Whenever we're doing anything algebraically it's always a little bit useful to see if we can isolate the variable that we're trying to solve for, we're trying to find what t value will make this equal that right over there. A good first step would maybe try to get this five out of the left hand side, so let's divide the left by five. If we want to keep this being in equality we have to do the same thing to both sides. We get two to the t power is equal to 1,111 over five. How do we solve for t here? What function is essentially the inverse of the exponential function? Well it would be the logarithm. If we say that a to the b power is equal to c then that means that log base a of c is equal to b. a to the b power is equal to c. Log base a of c says what power do I need to raise a to, to get to c? Well I need to raise a to the b power, to get to c. a to the b power is equal to c. These two are actually equivalent statements. Let's take log base two of both sides of this equation. On the left hand side you have log base two of two to the t power. On the right hand side, you have log base two of 1,111 over five. Why is this useful right over here? This is what power do we have to raise two to, to get two to t power? Well to get two to the t power, we have to raise two to the t power. This thing right over here just simplifies to t. That just simplifies to t. On the right hand side, we have log base two, we have all of these business right over here. I'll just write it over, t is equal to log base two of 1,111 over five. This is an expression that gives us our t value but then the next question is well how do we figure out what this is? If you take out your calculator, you'll quickly notice that there is no log base two button, so how do we actually compute it? Here we just have to apply a very useful property of exponents. If we have log base two of well really anything. Let me write it this way, if we have log base a of c, we can compute this as log base anything of c over log base that same anything of a. This anything has to be the same thing. Our calculator is useful because it has a log, you just press log, it's log base 10. If you press ln, it's natural log or log base e. I like to just use the log base 10, so this is going to be the same thing as log base 10 of 1,111 over five over log base 10 of two. We can get our calculator out and we could unlog base e if we wanted that would be a natural log but I'll just use the log button. This is logarithm of 1,111 over five, so that's this part right over here. This is log base 10, implicitly that's what the log button is. Divided by log base 10 of two and then that gives us seven, well it just keeps on going but is approximately equal to 7.796. This is approximately equal to 7.796, so when t is roughly equal to that you're going to have y equaling 1,111.