<div>
<p>Given a equation <img title="tex_920788608923e1a29dcc453bcaa190e7 Find the Real Root of 4^x + 6^x = 9^x math " alt="tex_920788608923e1a29dcc453bcaa190e7 Find the Real Root of 4^x + 6^x = 9^x math " src='https://helloacm.com/wp-content/uploads/easy-latex-cache/tex_920788608923e1a29dcc453bcaa190e7.png' title='4^x + 6^x = 9^x ' style='vertical-align:1%' class='tex' alt='4^x + 6^x = 9^x ' />, let&#8217;s find the real value of x.</p><p>First, divide both sides by <img title="tex_3713d39bf3977004053c53c30e5c84e4 Find the Real Root of 4^x + 6^x = 9^x math " alt="tex_3713d39bf3977004053c53c30e5c84e4 Find the Real Root of 4^x + 6^x = 9^x math " src='https://helloacm.com/wp-content/uploads/easy-latex-cache/tex_3713d39bf3977004053c53c30e5c84e4.png' title='4^x ' style='vertical-align:1%' class='tex' alt='4^x ' /> which becomes:</p><p><img title="tex_85d895cc0e7faade1d6b803a6d2d94f5 Find the Real Root of 4^x + 6^x = 9^x math " alt="tex_85d895cc0e7faade1d6b803a6d2d94f5 Find the Real Root of 4^x + 6^x = 9^x math " src='https://helloacm.com/wp-content/uploads/easy-latex-cache/tex_85d895cc0e7faade1d6b803a6d2d94f5.png' title='1 + \frac{6^x}{4^x} = \frac{9^x}{4^x} ' style='vertical-align:1%' class='tex' alt='1 + \frac{6^x}{4^x} = \frac{9^x}{4^x} ' /><br /> as <img title="tex_1ac70cc9e9bf75d0f3cdc56625c5d08d Find the Real Root of 4^x + 6^x = 9^x math " alt="tex_1ac70cc9e9bf75d0f3cdc56625c5d08d Find the Real Root of 4^x + 6^x = 9^x math " src='https://helloacm.com/wp-content/uploads/easy-latex-cache/tex_1ac70cc9e9bf75d0f3cdc56625c5d08d.png' title='6/4 = 3/2 ' style='vertical-align:1%' class='tex' alt='6/4 = 3/2 ' />, and <img title="tex_b3d6ca55e09f520d99de7976e24efaba Find the Real Root of 4^x + 6^x = 9^x math " alt="tex_b3d6ca55e09f520d99de7976e24efaba Find the Real Root of 4^x + 6^x = 9^x math " src='https://helloacm.com/wp-content/uploads/easy-latex-cache/tex_b3d6ca55e09f520d99de7976e24efaba.png' title='9/4 = (3/2)^2 ' style='vertical-align:1%' class='tex' alt='9/4 = (3/2)^2 ' />, it becomes:</p><p><img title="tex_6acc2c885bd3873669b5aeada000a14a Find the Real Root of 4^x + 6^x = 9^x math " alt="tex_6acc2c885bd3873669b5aeada000a14a Find the Real Root of 4^x + 6^x = 9^x math " src='https://helloacm.com/wp-content/uploads/easy-latex-cache/tex_6acc2c885bd3873669b5aeada000a14a.png' title='1 + (\frac{3}{2}) ^x = ((\frac{3}{2})^2)^x = ((\frac{3}{2}))^2x ' style='vertical-align:1%' class='tex' alt='1 + (\frac{3}{2}) ^x = ((\frac{3}{2})^2)^x = ((\frac{3}{2}))^2x ' /></p><p>Let&#8217;s replace <img title="tex_ab7c8bd1f7fbb973aa58d541a7f65f78 Find the Real Root of 4^x + 6^x = 9^x math " alt="tex_ab7c8bd1f7fbb973aa58d541a7f65f78 Find the Real Root of 4^x + 6^x = 9^x math " src='https://helloacm.com/wp-content/uploads/easy-latex-cache/tex_ab7c8bd1f7fbb973aa58d541a7f65f78.png' title='(\frac{3}{2}) ^x ' style='vertical-align:1%' class='tex' alt='(\frac{3}{2}) ^x ' /> with <img title="tex_08d76dd2192e1355ed65b2f013af4415 Find the Real Root of 4^x + 6^x = 9^x math " alt="tex_08d76dd2192e1355ed65b2f013af4415 Find the Real Root of 4^x + 6^x = 9^x math " src='https://helloacm.com/wp-content/uploads/easy-latex-cache/tex_08d76dd2192e1355ed65b2f013af4415.png' title='u ' style='vertical-align:1%' class='tex' alt='u ' /> and further we have:</p><p><img title="tex_d576ee8ed6e66bc97dfbdb0e3c923598 Find the Real Root of 4^x + 6^x = 9^x math " alt="tex_d576ee8ed6e66bc97dfbdb0e3c923598 Find the Real Root of 4^x + 6^x = 9^x math " src='https://helloacm.com/wp-content/uploads/easy-latex-cache/tex_d576ee8ed6e66bc97dfbdb0e3c923598.png' title='1 + u = u^2 ' style='vertical-align:1%' class='tex' alt='1 + u = u^2 ' /> that is <img title="tex_dcdf980b45bf9707b52e3d2ac74fb84f Find the Real Root of 4^x + 6^x = 9^x math " alt="tex_dcdf980b45bf9707b52e3d2ac74fb84f Find the Real Root of 4^x + 6^x = 9^x math " src='https://helloacm.com/wp-content/uploads/easy-latex-cache/tex_dcdf980b45bf9707b52e3d2ac74fb84f.png' title='u^2 - u - 1 = 0 ' style='vertical-align:1%' class='tex' alt='u^2 - u - 1 = 0 ' /></p><p>This equation has a golden root (the negative root should be discarded):</p><p><img title="tex_84f336b54e94a3f6a1141784af0afa4f Find the Real Root of 4^x + 6^x = 9^x math " alt="tex_84f336b54e94a3f6a1141784af0afa4f Find the Real Root of 4^x + 6^x = 9^x math " src='https://helloacm.com/wp-content/uploads/easy-latex-cache/tex_84f336b54e94a3f6a1141784af0afa4f.png' title='u = \frac{1 + \sqrt{5}}{2} ' style='vertical-align:1%' class='tex' alt='u = \frac{1 + \sqrt{5}}{2} ' /></p><p>Therefore, <img title="tex_6a0d8158137223f60381ab35de616748 Find the Real Root of 4^x + 6^x = 9^x math " alt="tex_6a0d8158137223f60381ab35de616748 Find the Real Root of 4^x + 6^x = 9^x math " src='https://helloacm.com/wp-content/uploads/easy-latex-cache/tex_6a0d8158137223f60381ab35de616748.png' title='(\frac{3}{2}) ^x = \frac{1 + \sqrt{5}}{2} ' style='vertical-align:1%' class='tex' alt='(\frac{3}{2}) ^x = \frac{1 + \sqrt{5}}{2} ' /><br /> and <img title="tex_57c96216012ab243d3db15aa5ddff79c Find the Real Root of 4^x + 6^x = 9^x math " alt="tex_57c96216012ab243d3db15aa5ddff79c Find the Real Root of 4^x + 6^x = 9^x math " src='https://helloacm.com/wp-content/uploads/easy-latex-cache/tex_57c96216012ab243d3db15aa5ddff79c.png' title='x = \log_{\frac{3}{2}}^{\frac{1 + \sqrt{5}}{2}} ' style='vertical-align:1%' class='tex' alt='x = \log_{\frac{3}{2}}^{\frac{1 + \sqrt{5}}{2}} ' /><br /> and <img title="tex_bb249fb49321758ad7d8db33e6550647 Find the Real Root of 4^x + 6^x = 9^x math " alt="tex_bb249fb49321758ad7d8db33e6550647 Find the Real Root of 4^x + 6^x = 9^x math " src='https://helloacm.com/wp-content/uploads/easy-latex-cache/tex_bb249fb49321758ad7d8db33e6550647.png' title='x = \frac{\ln{\frac{1 + \sqrt{5}}{2}}}{\ln{\frac{3}{2}}} ' style='vertical-align:1%' class='tex' alt='x = \frac{\ln{\frac{1 + \sqrt{5}}{2}}}{\ln{\frac{3}{2}}} ' /><br /> and we can compute the value x that is approximately 1.187.</p>
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--EOF (<a title='The Ultimate Computing & Technology Blog' href='https://helloacm.com'>The Ultimate Computing & Technology Blog</a>) -- 

Reposted to [Blog](https://helloacm.com/find-the-real-root-of-4x-6x-9x/)
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