12th Grade
What is the Inverse Relationship Between Exponents and Logarithms?
{
"voice_prompt": "",
"manuscript": {
"title": {
"text": "What is the Inverse Relationship Between Exponents and Logarithms?",
"audio": "What is the Inverse Relationship Between Exponents and Logarithms?"
},
"description": {
"text": "Exponents and logarithms are opposites: one undoes the other. While exponents ask - What do you get when you raise a number to a power?,- logarithms ask - What power did you raise the number to in order to get this result?",
"audio": "Exponents and logarithms are opposites: one undoes the other. While exponents ask - what do you get when you raise a number to a power?,- logarithms ask - what power did you raise the number to in order to get this result?"
},
"scenes": [
{
"text": "Exponents and logarithms are inverse operations. That means one undoes the effect of the other. If you raise a base b to the power x and get y, then taking the logarithm base b of y brings you back to x. This relation is useful to know when solving both exponential and logarithmic equations",
"latex": "b^x = y \\quad \\implies \\quad \\log_b(y) = x"
},
{
"text": "Let's look at a specific example. Recall that ten squared equals one hundred. If you take the logarithm with base ten of one hundred, you get two. This shows how logarithms reverse exponentiation.",
"latex": "10^2 = 100 \\quad \\implies \\quad \\log_{10}(100) = 2"
},
{
"text": "This inverse relationship works both ways. If you start with the logarithm with base two of eight equals three, then that tells you two cubed equals eight.",
"latex": "\\log_2(8) = 3 \\quad \\implies \\quad 2^3 = 8"
},
{
"text": "Here's another example: five to the fourth equals six hundred twenty-five, so the logarithm with base five of six hundred twenty-five equals four. This shows the base-five logarithm brings us back to the exponent.",
"latex": "5^4 = 625 \\quad \\implies \\quad \\log_5(625) = 4"
},
{
"text": "In general, for any positive base b not equal to one, the logarithm of b to the x is x, and b raised to the logarithm with base b of y is y.",
"latex": "\\log_b(b^x) = x, \\quad b^{\\log_b(y)} = y"
},
{
"text": "This inverse relationship is useful when solving exponential and logarithmic equations. For example, to solve three to the x equals eighty-one, you can take the logarithm base three of both sides to get x equals the logarithm with base three of eighty-one, which equals four.",
"latex": "3^x = 81 \\quad \\implies \\quad x = \\log_3(81) = 4"
},
{
"text": "Let's apply this in real life. In sound, the decibel scale is logarithmic. If sound intensity increases by a factor of ten, the decibel level increases by ten. This is because decibel equals ten times the logarithm with base ten of the intensity ratio.",
"latex": "\\text{dB} = 10 \\times \\log_{10}(\\frac{\\text{I}}{\\text{I}_0})"
},
],
"outro": {
"text": "Exponents and logarithms undo each other. This inverse relationship helps us solve equations and understand how exponential growth or decay connects to logarithmic scales — from sound levels to pH in chemistry.",
"audio": "Exponents and logarithms undo each other. This inverse relationship helps us solve equations and understand how exponential growth or decay connects to logarithmic scales — from sound levels to pH in chemistry."
}
}
}
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