1984 AIME Problems/Problem 5: Difference between revisions

Revision as of 21:42, 6 March 2025

Problem

Determine the value of $ab$ if $\log_8a+\log_4b^2=5$ and $\log_8b+\log_4a^2=7$ .

Solution 1

Use the change of base formula to see that $\frac{\log a}{\log 8} + \frac{2 \log b}{\log 4} = 5$ ; combine denominators to find that $\frac{\log ab^3}{3\log 2} = 5$ . Doing the same thing with the second equation yields that $\frac{\log a^3b}{3\log 2} = 7$ . This means that $\log ab^3 = 15\log 2 \Longrightarrow ab^3 = 2^{15}$ and that $\log a^3 b = 21\log 2 \Longrightarrow a^3 b = 2^{21}$ . If we multiply the two equations together, we get that $a^4b^4 = 2^{36}$ , so taking the fourth root of that, $ab = 2^9 = \boxed{512}$ .

Solution 2

We can simplify our expressions by changing everything to a common base and by pulling exponents out of the logarithms. The given equations then become $\frac{\ln a}{\ln 8} + \frac{2 \ln b}{\ln 4} = 5$ and $\frac{\ln b}{\ln 8} + \frac{2 \ln a}{\ln 4} = 7$ . Adding the equations and factoring, we get $(\frac{1}{\ln 8}+\frac{2}{\ln 4})(\ln a+ \ln b)=12$ . Rearranging we see that $\ln ab = \frac{12}{\frac{1}{\ln 8}+\frac{2}{\ln 4}}$ . Again, we pull exponents out of our logarithms to get $\ln ab = \frac{12}{\frac{1}{3 \ln 2} + \frac{2}{2 \ln 2}} = \frac{12 \ln 2}{\frac{1}{3} + 1} = \frac{12 \ln 2}{\frac{4}{3}} = 9 \ln 2$ . This means that $\frac{\ln ab}{\ln 2} = 9$ . The left-hand side can be interpreted as a base-2 logarithm, giving us $ab = 2^9 = \boxed{512}$ .

Solution 3

This solution is very similar to the above two, but it utilizes the well-known fact that $\log_{m^k}{n^k}= \log_m{n}.$ Thus, $\log_8a+\log_4b^2=5 \Rightarrow \log_{2^3}{(\sqrt[3]{a})^3} + \log_{2^2}{b^2} = 5 \Rightarrow \log_2{\sqrt[3]{a}} + \log_2{b} = 5 \Rightarrow \log_2{\sqrt[3]{a}b} = 5.$ Similarly, $\log_8b+\log_4a^2=7 \Rightarrow \log_2{\sqrt[3]{b}a} = 7.$ Adding these two equations, we have $\log_2{a^{\frac{4}{3}}b^{\frac{4}{3}}} = 12 \Rightarrow ab = 2^{12\times\frac{3}{4}} = 2^9 = \boxed{512}$ .

Solution 4

We can change everything to a common base, like so: $\log_8{a} + \log_8{b^3} = 5,$ $\log_8{b} + \log_8{a^3} = 7.$ We set the value of $\log_8{a}$ to $x$ , and the value of $\log_8{b}$ to $y.$ Now we have a system of linear equations: $x + 3y = 5,$ $y + 3x = 7.$ Now add the two equations together then simplify, we'll get $x+y=3$ . So $\log_8{ab} = \log_8{a} + \log_8{b} = 3$ , $ab = 8^3 = \boxed{512}$

Solution 5

Add the two equations to get $\log_8 {a}+ \log_8 {b}+ \log_{a^2}+\log_{b^2}=12$ . This can be simplified with the log property $\log_n {x}+\log_n {y}=log_n {xy}$ . Using this, we get $\log_8 {ab}+ \log_4 {a^2b^2}=12$ . Now let $\log_8 {ab}=c$ and $\log_4 {a^2b^2}=k$ . Converting to exponents, we get $8^c=ab$ and $4^k=(ab)^2$ . Sub in the $8^c$ to get $k=3c$ . So now we have that $k+c=12$ and $k=3c$ which gives $c=3$ , $k=9$ . This means $\log_4 {a^2b^2}=9$ so $4^9=(ab)^2 \implies ab=(2^2)^9 \implies 2^9 \implies \boxed {512}$

Solution 6

Add the equations and use the facts that $\log{a} +\log{b}=\log{ab}$ and $\log{k^n} =n\log{k}$ to get $\log_8{ab} +2\log_4{ab}=12$ Now use the change of base identity with base as 2: $\dfrac{\log_2{ab}}{\log_2{8}}+\dfrac{2\log_2{ab}}{\log_2{4}}=12$ Which gives: $\frac{4}{3}\log_2{ab}=12$ Solving gives, $\boxed{ab=2^9=512}$

Solution 7

By properties of logarithms, we know that $\log_8 {a}+ \log_4 {b ^ 2} = \log_2 {a ^ {1/3}}+ \log_2 {b} = 5$ .

Using the fact that $\log_a {b} + \log_a {c} = log_a{b*c}$ , we get $\log_2 {a^{1/3} * b} = 5$ .

Similarly, we know that $\log_2 {a * b^{1/3}} = 7$ .

From these two equations, we get $a^{1/3} * b = 2^5$ and $a * b^{1/3} = 2^7$ .

Multiply the two equations to get $a^{4/3} * b^{4/3} = 2^{12}$ . Solving, we get that $a*b = 2^{12*3/4} = 2^9 =$ $\boxed{512}$ .

Solution 8

Adding both of the equations, we get $\log_8{ab} +2\log_4{ab}=12$ Furthermore, we see that $\log_4 {ab}$ is $\frac{3}{2}$ times $\log_8 {ab}.$ Substituting $\log_8 {ab}$ as $x,$ we get $x+3x=12,$ so $x=3.$ Therefore, we have $\log_8 {ab} = 3,$ so $ab= 8^3=\boxed{512}$ ~ math_comb01

Solution 9

Change all equations to base 64. We then get: $\log_{64}(a^2) + \log_{64}(b^6) = 5$ and $\log_{64}(b^2) + \log_{64}(a^6) = 7.$

Using the property $\log(a) + \log(b) = \log(ab)$, we get: $\log_{64}(a^2b^6) = 5$ and $\log_{64}(a^6b^2) = 7.$

Then: $a^2b^6 = 64^5$ and $a^6b^2 = 64^7.$

Simplifying, we have: $ab^3 = 8^5$ and $a^3b = 8^7.$

Substituting and solving, we get: $a = 8^2$ and $b = 8.$

Then: $ab = 8^3 = 512.$

Solution 10

Given: - $\log_8 a + \log_4 b^2 = 5$ - $\log_8 b + \log_4 a^2 = 7$

We set up a system by subtracting 2 from the second equation to make both equal to 5: - $\log_8 a + \log_4 b^2 = 5$ - $\log_8 b + \log_4 a^2 - 2 = 5$

Setting these equal: $\log_8 a + \log_4 b^2 = \log_8 b + \log_4 a^2 - 2$

Using the property $\log_4 x^2 = 2\log_4 x$ : $\log_8 a + 2\log_4 b = \log_8 b + 2\log_4 a - 2$

Converting to base 2, where $\log_8 x = \frac{\log_2 x}{3}$ and $\log_4 x = \frac{\log_2 x}{2}$ : $\frac{\log_2 a}{3} + 2\cdot\frac{\log_2 b}{2} = \frac{\log_2 b}{3} + 2\cdot\frac{\log_2 a}{2} - 2$

Simplifying: $\frac{\log_2 a}{3} + \log_2 b = \frac{\log_2 b}{3} + \log_2 a - 2$

Multiplying through by 6 to eliminate fractions: $2\log_2 a + 6\log_2 b = 2\log_2 b + 6\log_2 a - 12$

Rearranging: $2\log_2 a - 6\log_2 a + 6\log_2 b - 2\log_2 b = -12$ $-4\log_2 a + 4\log_2 b = -12$ $\log_2 b - \log_2 a = -3$

Therefore: $\log_2\left(\frac{b}{a}\right) = -3$ $\frac{b}{a} = \frac{1}{8}$ $b = \frac{a}{8}$

From the first original equation: $\log_8 a + \log_4 b^2 = 5$

Substituting $b = \frac{a}{8}$ : $\log_8 a + \log_4 \left(\frac{a}{8}\right)^2 = 5$ $\log_8 a + \log_4 \left(\frac{a^2}{64}\right) = 5$ $\log_8 a + \log_4 a^2 - \log_4 64 = 5$

Since $\log_4 64 = \log_4 4^3 = 3$ : $\log_8 a + \log_4 a^2 - 3 = 5$ $\log_8 a + \log_4 a^2 = 8$

Now we have: - $\log_8 a + \log_4 a^2 = 8$ (derived) - $\log_8 b + \log_4 a^2 = 7$ (given)

Subtracting: $\log_8 a - \log_8 b = 1$ $\log_8\left(\frac{a}{b}\right) = 1$ $\frac{a}{b} = 8$

This confirms our earlier finding that $b = \frac{a}{8}$ , so $a = 8b$ .

Substituting this back into the first original equation: $\log_8 (8b) + \log_4 b^2 = 5$ $\log_8 8 + \log_8 b + \log_4 b^2 = 5$ $1 + \log_8 b + \log_4 b^2 = 5$ $\log_8 b + \log_4 b^2 = 4$

From the second original equation: $\log_8 b + \log_4 (8b)^2 = 7$ $\log_8 b + \log_4 (64b^2) = 7$ $\log_8 b + \log_4 64 + \log_4 b^2 = 7$ $\log_8 b + 3 + \log_4 b^2 = 7$ $\log_8 b + \log_4 b^2 = 4$

Thus both equations yield the same constraint: $\log_8 b + \log_4 b^2 = 4$

Converting to base 2: $\frac{\log_2 b}{3} + \frac{2\log_2 b}{2} = 4$ $\frac{\log_2 b}{3} + \log_2 b = 4$ $\frac{\log_2 b + 3\log_2 b}{3} = 4$ $\frac{4\log_2 b}{3} = 4$ $\log_2 b = 3$

Therefore: $b = 2^3 = 8$ $a = 8b = 8 \cdot 8 = 64$ $ab = 64 \cdot 8 = 512$

The value of $ab$ is $\boxed{512}$ .

~ brandonyee

@@ Line 72: / Line 72: @@
 Then:
 <cmath> ab = 8^3 = 512. </cmath>
+== Solution 10 ==
+Given:
+- <math>\log_8 a + \log_4 b^2 = 5</math>
+- <math>\log_8 b + \log_4 a^2 = 7</math>
+We set up a system by subtracting 2 from the second equation to make both equal to 5:
+- <math>\log_8 a + \log_4 b^2 = 5</math>
+- <math>\log_8 b + \log_4 a^2 - 2 = 5</math>
+Setting these equal:
+<math>\log_8 a + \log_4 b^2 = \log_8 b + \log_4 a^2 - 2</math>
+Using the property <math>\log_4 x^2 = 2\log_4 x</math>:
+<math>\log_8 a + 2\log_4 b = \log_8 b + 2\log_4 a - 2</math>
+Converting to base 2, where <math>\log_8 x = \frac{\log_2 x}{3}</math> and <math>\log_4 x = \frac{\log_2 x}{2}</math>:
+<math>\frac{\log_2 a}{3} + 2\cdot\frac{\log_2 b}{2} = \frac{\log_2 b}{3} + 2\cdot\frac{\log_2 a}{2} - 2</math>
+Simplifying:
+<math>\frac{\log_2 a}{3} + \log_2 b = \frac{\log_2 b}{3} + \log_2 a - 2</math>
+Multiplying through by 6 to eliminate fractions:
+<math>2\log_2 a + 6\log_2 b = 2\log_2 b + 6\log_2 a - 12</math>
+Rearranging:
+<math>2\log_2 a - 6\log_2 a + 6\log_2 b - 2\log_2 b = -12</math>
+<math>-4\log_2 a + 4\log_2 b = -12</math>
+<math>\log_2 b - \log_2 a = -3</math>
+Therefore:
+<math>\log_2\left(\frac{b}{a}\right) = -3</math>
+<math>\frac{b}{a} = \frac{1}{8}</math>
+<math>b = \frac{a}{8}</math>
+From the first original equation:
+<math>\log_8 a + \log_4 b^2 = 5</math>
+Substituting <math>b = \frac{a}{8}</math>:
+<math>\log_8 a + \log_4 \left(\frac{a}{8}\right)^2 = 5</math>
+<math>\log_8 a + \log_4 \left(\frac{a^2}{64}\right) = 5</math>
+<math>\log_8 a + \log_4 a^2 - \log_4 64 = 5</math>
+Since <math>\log_4 64 = \log_4 4^3 = 3</math>:
+<math>\log_8 a + \log_4 a^2 - 3 = 5</math>
+<math>\log_8 a + \log_4 a^2 = 8</math>
+Now we have:
+- <math>\log_8 a + \log_4 a^2 = 8</math> (derived)
+- <math>\log_8 b + \log_4 a^2 = 7</math> (given)
+Subtracting:
+<math>\log_8 a - \log_8 b = 1</math>
+<math>\log_8\left(\frac{a}{b}\right) = 1</math>
+<math>\frac{a}{b} = 8</math>
+This confirms our earlier finding that <math>b = \frac{a}{8}</math>, so <math>a = 8b</math>.
+Substituting this back into the first original equation:
+<math>\log_8 (8b) + \log_4 b^2 = 5</math>
+<math>\log_8 8 + \log_8 b + \log_4 b^2 = 5</math>
+<math>1 + \log_8 b + \log_4 b^2 = 5</math>
+<math>\log_8 b + \log_4 b^2 = 4</math>
+From the second original equation:
+<math>\log_8 b + \log_4 (8b)^2 = 7</math>
+<math>\log_8 b + \log_4 (64b^2) = 7</math>
+<math>\log_8 b + \log_4 64 + \log_4 b^2 = 7</math>
+<math>\log_8 b + 3 + \log_4 b^2 = 7</math>
+<math>\log_8 b + \log_4 b^2 = 4</math>
+Thus both equations yield the same constraint: <math>\log_8 b + \log_4 b^2 = 4</math>
+Converting to base 2:
+<math>\frac{\log_2 b}{3} + \frac{2\log_2 b}{2} = 4</math>
+<math>\frac{\log_2 b}{3} + \log_2 b = 4</math>
+<math>\frac{\log_2 b + 3\log_2 b}{3} = 4</math>
+<math>\frac{4\log_2 b}{3} = 4</math>
+<math>\log_2 b = 3</math>
+Therefore:
+<math>b = 2^3 = 8</math>
+<math>a = 8b = 8 \cdot 8 = 64</math>
+<math>ab = 64 \cdot 8 = 512</math>
+The value of <math>ab</math> is <math>\boxed{512}</math>.
+~ brandonyee
 == See also ==

1984 AIME (Problems • Answer Key • Resources)
Preceded by Problem 4	Followed by Problem 6
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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